:: MESFUNC9 semantic presentation begin theorem Th1: :: MESFUNC9:1 for X being non empty set for f, g being PartFunc of X,ExtREAL st f is V120() & g is V120() holds dom (f + g) = (dom f) /\ (dom g) proof let X be non empty set ; ::_thesis: for f, g being PartFunc of X,ExtREAL st f is V120() & g is V120() holds dom (f + g) = (dom f) /\ (dom g) let f, g be PartFunc of X,ExtREAL; ::_thesis: ( f is V120() & g is V120() implies dom (f + g) = (dom f) /\ (dom g) ) assume that A1: f is V120() and A2: g is V120() ; ::_thesis: dom (f + g) = (dom f) /\ (dom g) not +infty in rng g by A2, MESFUNC5:def_4; then A3: g " {+infty} = {} by FUNCT_1:72; not +infty in rng f by A1, MESFUNC5:def_4; then f " {+infty} = {} by FUNCT_1:72; then ((f " {+infty}) /\ (g " {-infty})) \/ ((f " {-infty}) /\ (g " {+infty})) = {} by A3; then dom (f + g) = ((dom f) /\ (dom g)) \ {} by MESFUNC1:def_3; hence dom (f + g) = (dom f) /\ (dom g) ; ::_thesis: verum end; theorem Th2: :: MESFUNC9:2 for X being non empty set for f, g being PartFunc of X,ExtREAL st f is V120() & g is V119() holds dom (f - g) = (dom f) /\ (dom g) proof let X be non empty set ; ::_thesis: for f, g being PartFunc of X,ExtREAL st f is V120() & g is V119() holds dom (f - g) = (dom f) /\ (dom g) let f, g be PartFunc of X,ExtREAL; ::_thesis: ( f is V120() & g is V119() implies dom (f - g) = (dom f) /\ (dom g) ) assume that A1: f is V120() and A2: g is V119() ; ::_thesis: dom (f - g) = (dom f) /\ (dom g) not +infty in rng f by A1, MESFUNC5:def_4; then A3: f " {+infty} = {} by FUNCT_1:72; not -infty in rng g by A2, MESFUNC5:def_3; then g " {-infty} = {} by FUNCT_1:72; then ((g " {+infty}) /\ (f " {+infty})) \/ ((g " {-infty}) /\ (f " {-infty})) = {} by A3; then dom (f - g) = ((dom f) /\ (dom g)) \ {} by MESFUNC1:def_4; hence dom (f - g) = (dom f) /\ (dom g) ; ::_thesis: verum end; theorem Th3: :: MESFUNC9:3 for X being non empty set for f, g being PartFunc of X,ExtREAL st f is V119() & g is V119() holds f + g is V119() proof let X be non empty set ; ::_thesis: for f, g being PartFunc of X,ExtREAL st f is V119() & g is V119() holds f + g is V119() let f, g be PartFunc of X,ExtREAL; ::_thesis: ( f is V119() & g is V119() implies f + g is V119() ) assume that A1: f is V119() and A2: g is V119() ; ::_thesis: f + g is V119() A3: dom (f + g) = (dom f) /\ (dom g) by A1, A2, MESFUNC5:16; for x being set st x in dom (f + g) holds -infty < (f + g) . x proof let x be set ; ::_thesis: ( x in dom (f + g) implies -infty < (f + g) . x ) assume A4: x in dom (f + g) ; ::_thesis: -infty < (f + g) . x then x in dom f by A3, XBOOLE_0:def_4; then A5: -infty < f . x by A1, MESFUNC5:10; x in dom g by A3, A4, XBOOLE_0:def_4; then A6: -infty < g . x by A2, MESFUNC5:10; (f + g) . x = (f . x) + (g . x) by A4, MESFUNC1:def_3; hence -infty < (f + g) . x by A5, A6, XXREAL_0:6, XXREAL_3:17; ::_thesis: verum end; hence f + g is V119() by MESFUNC5:10; ::_thesis: verum end; theorem Th4: :: MESFUNC9:4 for X being non empty set for f, g being PartFunc of X,ExtREAL st f is V120() & g is V120() holds f + g is V120() proof let X be non empty set ; ::_thesis: for f, g being PartFunc of X,ExtREAL st f is V120() & g is V120() holds f + g is V120() let f, g be PartFunc of X,ExtREAL; ::_thesis: ( f is V120() & g is V120() implies f + g is V120() ) assume that A1: f is V120() and A2: g is V120() ; ::_thesis: f + g is V120() A3: dom (f + g) = (dom f) /\ (dom g) by A1, A2, Th1; for x being set st x in dom (f + g) holds (f + g) . x < +infty proof let x be set ; ::_thesis: ( x in dom (f + g) implies (f + g) . x < +infty ) assume A4: x in dom (f + g) ; ::_thesis: (f + g) . x < +infty then x in dom f by A3, XBOOLE_0:def_4; then A5: f . x < +infty by A1, MESFUNC5:11; x in dom g by A3, A4, XBOOLE_0:def_4; then A6: g . x < +infty by A2, MESFUNC5:11; (f + g) . x = (f . x) + (g . x) by A4, MESFUNC1:def_3; hence (f + g) . x < +infty by A5, A6, XXREAL_0:4, XXREAL_3:16; ::_thesis: verum end; hence f + g is V120() by MESFUNC5:11; ::_thesis: verum end; theorem :: MESFUNC9:5 for X being non empty set for f, g being PartFunc of X,ExtREAL st f is V119() & g is V120() holds f - g is V119() proof let X be non empty set ; ::_thesis: for f, g being PartFunc of X,ExtREAL st f is V119() & g is V120() holds f - g is V119() let f, g be PartFunc of X,ExtREAL; ::_thesis: ( f is V119() & g is V120() implies f - g is V119() ) assume that A1: f is V119() and A2: g is V120() ; ::_thesis: f - g is V119() A3: dom (f - g) = (dom f) /\ (dom g) by A1, A2, MESFUNC5:17; for x being set st x in dom (f - g) holds -infty < (f - g) . x proof let x be set ; ::_thesis: ( x in dom (f - g) implies -infty < (f - g) . x ) assume A4: x in dom (f - g) ; ::_thesis: -infty < (f - g) . x then x in dom f by A3, XBOOLE_0:def_4; then A5: -infty < f . x by A1, MESFUNC5:10; x in dom g by A3, A4, XBOOLE_0:def_4; then A6: g . x < +infty by A2, MESFUNC5:11; (f - g) . x = (f . x) - (g . x) by A4, MESFUNC1:def_4; hence -infty < (f - g) . x by A5, A6, XXREAL_0:6, XXREAL_3:19; ::_thesis: verum end; hence f - g is V119() by MESFUNC5:10; ::_thesis: verum end; theorem :: MESFUNC9:6 for X being non empty set for f, g being PartFunc of X,ExtREAL st f is V120() & g is V119() holds f - g is V120() proof let X be non empty set ; ::_thesis: for f, g being PartFunc of X,ExtREAL st f is V120() & g is V119() holds f - g is V120() let f, g be PartFunc of X,ExtREAL; ::_thesis: ( f is V120() & g is V119() implies f - g is V120() ) assume that A1: f is V120() and A2: g is V119() ; ::_thesis: f - g is V120() A3: dom (f - g) = (dom f) /\ (dom g) by A1, A2, Th2; for x being set st x in dom (f - g) holds (f - g) . x < +infty proof let x be set ; ::_thesis: ( x in dom (f - g) implies (f - g) . x < +infty ) assume A4: x in dom (f - g) ; ::_thesis: (f - g) . x < +infty then x in dom f by A3, XBOOLE_0:def_4; then A5: f . x < +infty by A1, MESFUNC5:11; x in dom g by A3, A4, XBOOLE_0:def_4; then A6: -infty < g . x by A2, MESFUNC5:10; (f - g) . x = (f . x) - (g . x) by A4, MESFUNC1:def_4; hence (f - g) . x < +infty by A5, A6, XXREAL_0:4, XXREAL_3:18; ::_thesis: verum end; hence f - g is V120() by MESFUNC5:11; ::_thesis: verum end; theorem Th7: :: MESFUNC9:7 for seq being ExtREAL_sequence holds ( ( seq is convergent_to_finite_number implies ex g being real number st ( lim seq = g & ( for p being real number st 0 < p holds ex n being Nat st for m being Nat st n <= m holds |.((seq . m) - (lim seq)).| < p ) ) ) & ( seq is convergent_to_+infty implies lim seq = +infty ) & ( seq is convergent_to_-infty implies lim seq = -infty ) ) proof let seq be ExtREAL_sequence; ::_thesis: ( ( seq is convergent_to_finite_number implies ex g being real number st ( lim seq = g & ( for p being real number st 0 < p holds ex n being Nat st for m being Nat st n <= m holds |.((seq . m) - (lim seq)).| < p ) ) ) & ( seq is convergent_to_+infty implies lim seq = +infty ) & ( seq is convergent_to_-infty implies lim seq = -infty ) ) A1: now__::_thesis:_(_seq_is_convergent_to_finite_number_implies_ex_g,_g_being_real_number_st_ (_lim_seq_=_g_&_(_for_p_being_real_number_st_0_<_p_holds_ ex_n_being_Nat_st_ for_m_being_Nat_st_n_<=_m_holds_ |.((seq_._m)_-_(lim_seq)).|_<_p_)_)_) assume A2: seq is convergent_to_finite_number ; ::_thesis: ex g, g being real number st ( lim seq = g & ( for p being real number st 0 < p holds ex n being Nat st for m being Nat st n <= m holds |.((seq . m) - (lim seq)).| < p ) ) then A3: not seq is convergent_to_+infty by MESFUNC5:50; A4: not seq is convergent_to_-infty by A2, MESFUNC5:51; seq is convergent by A2, MESFUNC5:def_11; then consider g being real number such that A5: lim seq = g and A6: for p being real number st 0 < p holds ex n being Nat st for m being Nat st n <= m holds |.((seq . m) - (lim seq)).| < p and seq is convergent_to_finite_number by A3, A4, MESFUNC5:def_12; take g = g; ::_thesis: ex g being real number st ( lim seq = g & ( for p being real number st 0 < p holds ex n being Nat st for m being Nat st n <= m holds |.((seq . m) - (lim seq)).| < p ) ) thus ex g being real number st ( lim seq = g & ( for p being real number st 0 < p holds ex n being Nat st for m being Nat st n <= m holds |.((seq . m) - (lim seq)).| < p ) ) by A5, A6; ::_thesis: verum end; A7: now__::_thesis:_(_seq_is_convergent_to_-infty_implies_lim_seq_=_-infty_) assume A8: seq is convergent_to_-infty ; ::_thesis: lim seq = -infty then seq is convergent by MESFUNC5:def_11; hence lim seq = -infty by A8, MESFUNC5:def_12; ::_thesis: verum end; now__::_thesis:_(_seq_is_convergent_to_+infty_implies_lim_seq_=_+infty_) assume A9: seq is convergent_to_+infty ; ::_thesis: lim seq = +infty then seq is convergent by MESFUNC5:def_11; hence lim seq = +infty by A9, MESFUNC5:def_12; ::_thesis: verum end; hence ( ( seq is convergent_to_finite_number implies ex g being real number st ( lim seq = g & ( for p being real number st 0 < p holds ex n being Nat st for m being Nat st n <= m holds |.((seq . m) - (lim seq)).| < p ) ) ) & ( seq is convergent_to_+infty implies lim seq = +infty ) & ( seq is convergent_to_-infty implies lim seq = -infty ) ) by A1, A7; ::_thesis: verum end; theorem Th8: :: MESFUNC9:8 for seq being ExtREAL_sequence st seq is V111() holds not seq is convergent_to_-infty proof let seq be ExtREAL_sequence; ::_thesis: ( seq is V111() implies not seq is convergent_to_-infty ) assume A1: seq is V111() ; ::_thesis: not seq is convergent_to_-infty assume seq is convergent_to_-infty ; ::_thesis: contradiction then consider n being Nat such that A2: for m being Nat st n <= m holds seq . m <= - 1 by MESFUNC5:def_10; seq . n <= - 1 by A2; hence contradiction by A1, SUPINF_2:51; ::_thesis: verum end; theorem Th9: :: MESFUNC9:9 for seq being ExtREAL_sequence for p being ext-real number st seq is convergent & ( for k being Nat holds seq . k <= p ) holds lim seq <= p proof let seq be ExtREAL_sequence; ::_thesis: for p being ext-real number st seq is convergent & ( for k being Nat holds seq . k <= p ) holds lim seq <= p let p be ext-real number ; ::_thesis: ( seq is convergent & ( for k being Nat holds seq . k <= p ) implies lim seq <= p ) assume that A1: seq is convergent and A2: for k being Nat holds seq . k <= p ; ::_thesis: lim seq <= p for y being ext-real number st y in rng seq holds y <= p proof let y be ext-real number ; ::_thesis: ( y in rng seq implies y <= p ) assume y in rng seq ; ::_thesis: y <= p then consider x being set such that A3: x in dom seq and A4: y = seq . x by FUNCT_1:def_3; reconsider x = x as Nat by A3; seq . x <= p by A2; hence y <= p by A4; ::_thesis: verum end; then A5: p is UpperBound of rng seq by XXREAL_2:def_1; reconsider SUPSEQ = superior_realsequence seq as Function of NAT,ExtREAL ; consider Y being non empty Subset of ExtREAL such that A6: Y = { (seq . k) where k is Element of NAT : 0 <= k } and A7: SUPSEQ . 0 = sup Y by RINFSUP2:def_7; now__::_thesis:_for_x_being_set_st_x_in_rng_seq_holds_ x_in_Y let x be set ; ::_thesis: ( x in rng seq implies x in Y ) assume x in rng seq ; ::_thesis: x in Y then consider k being set such that A8: k in dom seq and A9: x = seq . k by FUNCT_1:def_3; thus x in Y by A6, A8, A9; ::_thesis: verum end; then A10: rng seq c= Y by TARSKI:def_3; for n being Element of NAT holds inf SUPSEQ <= SUPSEQ . n proof let n be Element of NAT ; ::_thesis: inf SUPSEQ <= SUPSEQ . n NAT = dom SUPSEQ by FUNCT_2:def_1; then SUPSEQ . n in rng SUPSEQ by FUNCT_1:def_3; hence inf SUPSEQ <= SUPSEQ . n by XXREAL_2:3; ::_thesis: verum end; then A11: inf SUPSEQ <= SUPSEQ . 0 ; now__::_thesis:_for_x_being_set_st_x_in_Y_holds_ x_in_rng_seq let x be set ; ::_thesis: ( x in Y implies x in rng seq ) assume x in Y ; ::_thesis: x in rng seq then A12: ex k being Element of NAT st ( x = seq . k & 0 <= k ) by A6; dom seq = NAT by FUNCT_2:def_1; hence x in rng seq by A12, FUNCT_1:3; ::_thesis: verum end; then Y c= rng seq by TARSKI:def_3; then Y = rng seq by A10, XBOOLE_0:def_10; then (superior_realsequence seq) . 0 <= p by A5, A7, XXREAL_2:def_3; then lim_sup seq <= p by A11, XXREAL_0:2; hence lim seq <= p by A1, RINFSUP2:41; ::_thesis: verum end; theorem Th10: :: MESFUNC9:10 for seq being ExtREAL_sequence for p being ext-real number st seq is convergent & ( for k being Nat holds p <= seq . k ) holds p <= lim seq proof let seq be ExtREAL_sequence; ::_thesis: for p being ext-real number st seq is convergent & ( for k being Nat holds p <= seq . k ) holds p <= lim seq let p be ext-real number ; ::_thesis: ( seq is convergent & ( for k being Nat holds p <= seq . k ) implies p <= lim seq ) assume that A1: seq is convergent and A2: for k being Nat holds p <= seq . k ; ::_thesis: p <= lim seq for y being ext-real number st y in rng seq holds p <= y proof let y be ext-real number ; ::_thesis: ( y in rng seq implies p <= y ) assume y in rng seq ; ::_thesis: p <= y then consider x being set such that A3: x in dom seq and A4: y = seq . x by FUNCT_1:def_3; reconsider x = x as Nat by A3; seq . x >= p by A2; hence p <= y by A4; ::_thesis: verum end; then A5: p is LowerBound of rng seq by XXREAL_2:def_2; reconsider INFSEQ = inferior_realsequence seq as Function of NAT,ExtREAL ; consider Y being non empty Subset of ExtREAL such that A6: Y = { (seq . k) where k is Element of NAT : 0 <= k } and A7: INFSEQ . 0 = inf Y by RINFSUP2:def_6; now__::_thesis:_for_x_being_set_st_x_in_rng_seq_holds_ x_in_Y let x be set ; ::_thesis: ( x in rng seq implies x in Y ) assume x in rng seq ; ::_thesis: x in Y then consider k being set such that A8: k in dom seq and A9: x = seq . k by FUNCT_1:def_3; thus x in Y by A6, A8, A9; ::_thesis: verum end; then A10: rng seq c= Y by TARSKI:def_3; for n being Element of NAT holds sup INFSEQ >= INFSEQ . n proof let n be Element of NAT ; ::_thesis: sup INFSEQ >= INFSEQ . n NAT = dom INFSEQ by FUNCT_2:def_1; then INFSEQ . n in rng INFSEQ by FUNCT_1:def_3; hence sup INFSEQ >= INFSEQ . n by XXREAL_2:4; ::_thesis: verum end; then A11: sup INFSEQ >= INFSEQ . 0 ; now__::_thesis:_for_x_being_set_st_x_in_Y_holds_ x_in_rng_seq let x be set ; ::_thesis: ( x in Y implies x in rng seq ) assume x in Y ; ::_thesis: x in rng seq then A12: ex k being Element of NAT st ( x = seq . k & 0 <= k ) by A6; dom seq = NAT by FUNCT_2:def_1; hence x in rng seq by A12, FUNCT_1:3; ::_thesis: verum end; then Y c= rng seq by TARSKI:def_3; then Y = rng seq by A10, XBOOLE_0:def_10; then (inferior_realsequence seq) . 0 >= p by A5, A7, XXREAL_2:def_4; then lim_inf seq >= p by A11, XXREAL_0:2; hence p <= lim seq by A1, RINFSUP2:41; ::_thesis: verum end; theorem Th11: :: MESFUNC9:11 for seq1, seq2, seq being ExtREAL_sequence st seq1 is convergent & seq2 is convergent & seq1 is V111() & seq2 is V111() & ( for k being Nat holds seq . k = (seq1 . k) + (seq2 . k) ) holds ( seq is V111() & seq is convergent & lim seq = (lim seq1) + (lim seq2) ) proof let seq1, seq2, seq be ExtREAL_sequence; ::_thesis: ( seq1 is convergent & seq2 is convergent & seq1 is V111() & seq2 is V111() & ( for k being Nat holds seq . k = (seq1 . k) + (seq2 . k) ) implies ( seq is V111() & seq is convergent & lim seq = (lim seq1) + (lim seq2) ) ) assume that A1: seq1 is convergent and A2: seq2 is convergent and A3: seq1 is V111() and A4: seq2 is V111() and A5: for k being Nat holds seq . k = (seq1 . k) + (seq2 . k) ; ::_thesis: ( seq is V111() & seq is convergent & lim seq = (lim seq1) + (lim seq2) ) A6: not seq2 is convergent_to_-infty by A4, Th8; for n being set st n in dom seq holds 0. <= seq . n proof let n be set ; ::_thesis: ( n in dom seq implies 0. <= seq . n ) assume n in dom seq ; ::_thesis: 0. <= seq . n then reconsider n1 = n as Nat ; A7: 0 <= seq2 . n1 by A4, SUPINF_2:51; A8: seq . n1 = (seq1 . n1) + (seq2 . n1) by A5; 0 <= seq1 . n1 by A3, SUPINF_2:51; hence 0. <= seq . n by A7, A8; ::_thesis: verum end; hence seq is V111() by SUPINF_2:52; ::_thesis: ( seq is convergent & lim seq = (lim seq1) + (lim seq2) ) A9: not seq1 is convergent_to_-infty by A3, Th8; for n being Nat holds 0 <= seq2 . n by A4, SUPINF_2:51; then A10: lim seq2 <> -infty by A2, Th10; percases ( seq1 is convergent_to_+infty or seq1 is convergent_to_finite_number ) by A1, A9, MESFUNC5:def_11; supposeA11: seq1 is convergent_to_+infty ; ::_thesis: ( seq is convergent & lim seq = (lim seq1) + (lim seq2) ) for g being real number st 0 < g holds ex n being Nat st for m being Nat st n <= m holds g <= seq . m proof let g be real number ; ::_thesis: ( 0 < g implies ex n being Nat st for m being Nat st n <= m holds g <= seq . m ) assume 0 < g ; ::_thesis: ex n being Nat st for m being Nat st n <= m holds g <= seq . m then consider n being Nat such that A12: for m being Nat st n <= m holds g <= seq1 . m by A11, MESFUNC5:def_9; take n ; ::_thesis: for m being Nat st n <= m holds g <= seq . m let m be Nat; ::_thesis: ( n <= m implies g <= seq . m ) assume n <= m ; ::_thesis: g <= seq . m then A13: g <= seq1 . m by A12; 0 <= seq2 . m by A4, SUPINF_2:51; then (R_EAL g) + 0. <= (seq1 . m) + (seq2 . m) by A13, XXREAL_3:36; then g <= (seq1 . m) + (seq2 . m) by XXREAL_3:4; hence g <= seq . m by A5; ::_thesis: verum end; then A14: seq is convergent_to_+infty by MESFUNC5:def_9; hence seq is convergent by MESFUNC5:def_11; ::_thesis: lim seq = (lim seq1) + (lim seq2) then A15: lim seq = +infty by A14, MESFUNC5:def_12; lim seq1 = +infty by A1, A11, MESFUNC5:def_12; hence lim seq = (lim seq1) + (lim seq2) by A10, A15, XXREAL_3:def_2; ::_thesis: verum end; supposeA16: seq1 is convergent_to_finite_number ; ::_thesis: ( seq is convergent & lim seq = (lim seq1) + (lim seq2) ) then A17: not seq1 is convergent_to_-infty by MESFUNC5:51; not seq1 is convergent_to_+infty by A16, MESFUNC5:50; then consider g being real number such that A18: lim seq1 = g and A19: for p being real number st 0 < p holds ex n being Nat st for m being Nat st n <= m holds |.((seq1 . m) - (lim seq1)).| < p and seq1 is convergent_to_finite_number by A1, A17, MESFUNC5:def_12; percases ( seq2 is convergent_to_+infty or seq2 is convergent_to_finite_number ) by A2, A6, MESFUNC5:def_11; supposeA20: seq2 is convergent_to_+infty ; ::_thesis: ( seq is convergent & lim seq = (lim seq1) + (lim seq2) ) for g being real number st 0 < g holds ex n being Nat st for m being Nat st n <= m holds g <= seq . m proof let g be real number ; ::_thesis: ( 0 < g implies ex n being Nat st for m being Nat st n <= m holds g <= seq . m ) assume 0 < g ; ::_thesis: ex n being Nat st for m being Nat st n <= m holds g <= seq . m then consider n being Nat such that A21: for m being Nat st n <= m holds g <= seq2 . m by A20, MESFUNC5:def_9; take n ; ::_thesis: for m being Nat st n <= m holds g <= seq . m let m be Nat; ::_thesis: ( n <= m implies g <= seq . m ) assume n <= m ; ::_thesis: g <= seq . m then A22: g <= seq2 . m by A21; 0 <= seq1 . m by A3, SUPINF_2:51; then (R_EAL g) + 0. <= (seq1 . m) + (seq2 . m) by A22, XXREAL_3:36; then g <= (seq1 . m) + (seq2 . m) by XXREAL_3:4; hence g <= seq . m by A5; ::_thesis: verum end; then A23: seq is convergent_to_+infty by MESFUNC5:def_9; hence seq is convergent by MESFUNC5:def_11; ::_thesis: lim seq = (lim seq1) + (lim seq2) then A24: lim seq = +infty by A23, MESFUNC5:def_12; lim seq2 = +infty by A2, A20, MESFUNC5:def_12; hence lim seq = (lim seq1) + (lim seq2) by A18, A24, XXREAL_3:def_2; ::_thesis: verum end; supposeA25: seq2 is convergent_to_finite_number ; ::_thesis: ( seq is convergent & lim seq = (lim seq1) + (lim seq2) ) then A26: not seq2 is convergent_to_-infty by MESFUNC5:51; not seq2 is convergent_to_+infty by A25, MESFUNC5:50; then consider h being real number such that A27: lim seq2 = h and A28: for p being real number st 0 < p holds ex n being Nat st for m being Nat st n <= m holds |.((seq2 . m) - (lim seq2)).| < p and seq2 is convergent_to_finite_number by A2, A26, MESFUNC5:def_12; reconsider h9 = h, g9 = g as Real by XREAL_0:def_1; A29: for p being real number st 0 < p holds ex n being Nat st for m being Nat st n <= m holds |.((seq . m) - (R_EAL (g + h))).| < p proof let p be real number ; ::_thesis: ( 0 < p implies ex n being Nat st for m being Nat st n <= m holds |.((seq . m) - (R_EAL (g + h))).| < p ) A30: R_EAL h = h9 ; assume A31: 0 < p ; ::_thesis: ex n being Nat st for m being Nat st n <= m holds |.((seq . m) - (R_EAL (g + h))).| < p then consider n1 being Nat such that A32: for m being Nat st n1 <= m holds |.((seq1 . m) - (lim seq1)).| < p / 2 by A19; consider n2 being Nat such that A33: for m being Nat st n2 <= m holds |.((seq2 . m) - (lim seq2)).| < p / 2 by A28, A31; reconsider n19 = n1, n29 = n2 as Element of NAT by ORDINAL1:def_12; reconsider n = max (n19,n29) as Nat ; take n ; ::_thesis: for m being Nat st n <= m holds |.((seq . m) - (R_EAL (g + h))).| < p let m be Nat; ::_thesis: ( n <= m implies |.((seq . m) - (R_EAL (g + h))).| < p ) assume A34: n <= m ; ::_thesis: |.((seq . m) - (R_EAL (g + h))).| < p n2 <= n by XXREAL_0:25; then n2 <= m by A34, XXREAL_0:2; then A35: |.((seq2 . m) - (lim seq2)).| < p / 2 by A33; then |.((seq2 . m) - (lim seq2)).| < +infty by XXREAL_0:2, XXREAL_0:9; then A36: (seq2 . m) - (R_EAL h) in REAL by A27, EXTREAL2:30; n1 <= n by XXREAL_0:25; then n1 <= m by A34, XXREAL_0:2; then A37: |.((seq1 . m) - (lim seq1)).| < p / 2 by A32; then |.((seq1 . m) - (lim seq1)).| < +infty by XXREAL_0:2, XXREAL_0:9; then (seq1 . m) - (R_EAL g) in REAL by A18, EXTREAL2:30; then consider e1, e2 being Real such that A38: e1 = (seq1 . m) - (R_EAL g) and A39: e2 = (seq2 . m) - (R_EAL h) by A36; A40: |.((seq2 . m) - (R_EAL h)).| = |.e2.| by A39, EXTREAL2:1; A41: 0 <= seq2 . m by A4, SUPINF_2:51; then A42: (seq2 . m) - (R_EAL h) <> -infty by XXREAL_3:19; A43: 0 <= seq1 . m by A3, SUPINF_2:51; A44: |.((seq1 . m) - (R_EAL g)).| = |.e1.| by A38, EXTREAL2:1; then A45: |.((seq2 . m) - (R_EAL h)).| + |.((seq1 . m) - (R_EAL g)).| = |.e1.| + |.e2.| by A40, SUPINF_2:1; R_EAL g = g9 ; then R_EAL (g + h) = (R_EAL g) + (R_EAL h) by A30, SUPINF_2:1; then (seq . m) - (R_EAL (g + h)) = ((seq . m) - (R_EAL h)) - (R_EAL g) by XXREAL_3:31 .= (((seq1 . m) + (seq2 . m)) - (R_EAL h)) - (R_EAL g) by A5 .= ((seq1 . m) + ((seq2 . m) - (R_EAL h))) - (R_EAL g) by A43, A41, XXREAL_3:30 .= ((seq2 . m) - (R_EAL h)) + ((seq1 . m) - (R_EAL g)) by A43, A42, XXREAL_3:30 ; then A46: |.((seq . m) - (R_EAL (g + h))).| <= |.((seq2 . m) - (R_EAL h)).| + |.((seq1 . m) - (R_EAL g)).| by EXTREAL2:13; |.e1.| + |.e2.| < (p / 2) + (p / 2) by A18, A27, A37, A35, A44, A40, XREAL_1:8; hence |.((seq . m) - (R_EAL (g + h))).| < p by A46, A45, XXREAL_0:2; ::_thesis: verum end; then A47: seq is convergent_to_finite_number by MESFUNC5:def_8; hence seq is convergent by MESFUNC5:def_11; ::_thesis: lim seq = (lim seq1) + (lim seq2) then lim seq = g9 + h9 by A29, A47, MESFUNC5:def_12; hence lim seq = (lim seq1) + (lim seq2) by A18, A27, SUPINF_2:1; ::_thesis: verum end; end; end; end; end; theorem Th12: :: MESFUNC9:12 for X being non empty set for G, F being Functional_Sequence of X,ExtREAL for x being Element of X for D being set st ( for n being Nat holds G . n = (F . n) | D ) & x in D holds ( ( F # x is convergent_to_+infty implies G # x is convergent_to_+infty ) & ( F # x is convergent_to_-infty implies G # x is convergent_to_-infty ) & ( F # x is convergent_to_finite_number implies G # x is convergent_to_finite_number ) & ( F # x is convergent implies G # x is convergent ) ) proof let X be non empty set ; ::_thesis: for G, F being Functional_Sequence of X,ExtREAL for x being Element of X for D being set st ( for n being Nat holds G . n = (F . n) | D ) & x in D holds ( ( F # x is convergent_to_+infty implies G # x is convergent_to_+infty ) & ( F # x is convergent_to_-infty implies G # x is convergent_to_-infty ) & ( F # x is convergent_to_finite_number implies G # x is convergent_to_finite_number ) & ( F # x is convergent implies G # x is convergent ) ) let G, F be Functional_Sequence of X,ExtREAL; ::_thesis: for x being Element of X for D being set st ( for n being Nat holds G . n = (F . n) | D ) & x in D holds ( ( F # x is convergent_to_+infty implies G # x is convergent_to_+infty ) & ( F # x is convergent_to_-infty implies G # x is convergent_to_-infty ) & ( F # x is convergent_to_finite_number implies G # x is convergent_to_finite_number ) & ( F # x is convergent implies G # x is convergent ) ) let x be Element of X; ::_thesis: for D being set st ( for n being Nat holds G . n = (F . n) | D ) & x in D holds ( ( F # x is convergent_to_+infty implies G # x is convergent_to_+infty ) & ( F # x is convergent_to_-infty implies G # x is convergent_to_-infty ) & ( F # x is convergent_to_finite_number implies G # x is convergent_to_finite_number ) & ( F # x is convergent implies G # x is convergent ) ) let D be set ; ::_thesis: ( ( for n being Nat holds G . n = (F . n) | D ) & x in D implies ( ( F # x is convergent_to_+infty implies G # x is convergent_to_+infty ) & ( F # x is convergent_to_-infty implies G # x is convergent_to_-infty ) & ( F # x is convergent_to_finite_number implies G # x is convergent_to_finite_number ) & ( F # x is convergent implies G # x is convergent ) ) ) assume that A1: for n being Nat holds G . n = (F . n) | D and A2: x in D ; ::_thesis: ( ( F # x is convergent_to_+infty implies G # x is convergent_to_+infty ) & ( F # x is convergent_to_-infty implies G # x is convergent_to_-infty ) & ( F # x is convergent_to_finite_number implies G # x is convergent_to_finite_number ) & ( F # x is convergent implies G # x is convergent ) ) thus A3: ( F # x is convergent_to_+infty implies G # x is convergent_to_+infty ) ::_thesis: ( ( F # x is convergent_to_-infty implies G # x is convergent_to_-infty ) & ( F # x is convergent_to_finite_number implies G # x is convergent_to_finite_number ) & ( F # x is convergent implies G # x is convergent ) ) proof assume A4: F # x is convergent_to_+infty ; ::_thesis: G # x is convergent_to_+infty let g be real number ; :: according to MESFUNC5:def_9 ::_thesis: ( g <= 0 or ex b1 being set st for b2 being set holds ( not b1 <= b2 or g <= (G # x) . b2 ) ) assume 0 < g ; ::_thesis: ex b1 being set st for b2 being set holds ( not b1 <= b2 or g <= (G # x) . b2 ) then consider n being Nat such that A5: for m being Nat st n <= m holds g <= (F # x) . m by A4, MESFUNC5:def_9; take n ; ::_thesis: for b1 being set holds ( not n <= b1 or g <= (G # x) . b1 ) let m be Nat; ::_thesis: ( not n <= m or g <= (G # x) . m ) assume n <= m ; ::_thesis: g <= (G # x) . m then g <= (F # x) . m by A5; then g <= (F . m) . x by MESFUNC5:def_13; then g <= ((F . m) | D) . x by A2, FUNCT_1:49; then g <= (G . m) . x by A1; hence g <= (G # x) . m by MESFUNC5:def_13; ::_thesis: verum end; thus A6: ( F # x is convergent_to_-infty implies G # x is convergent_to_-infty ) ::_thesis: ( ( F # x is convergent_to_finite_number implies G # x is convergent_to_finite_number ) & ( F # x is convergent implies G # x is convergent ) ) proof assume A7: F # x is convergent_to_-infty ; ::_thesis: G # x is convergent_to_-infty let g be real number ; :: according to MESFUNC5:def_10 ::_thesis: ( 0 <= g or ex b1 being set st for b2 being set holds ( not b1 <= b2 or (G # x) . b2 <= g ) ) assume g < 0 ; ::_thesis: ex b1 being set st for b2 being set holds ( not b1 <= b2 or (G # x) . b2 <= g ) then consider n being Nat such that A8: for m being Nat st n <= m holds (F # x) . m <= g by A7, MESFUNC5:def_10; take n ; ::_thesis: for b1 being set holds ( not n <= b1 or (G # x) . b1 <= g ) let m be Nat; ::_thesis: ( not n <= m or (G # x) . m <= g ) assume n <= m ; ::_thesis: (G # x) . m <= g then (F # x) . m <= g by A8; then (F . m) . x <= g by MESFUNC5:def_13; then ((F . m) | D) . x <= g by A2, FUNCT_1:49; then (G . m) . x <= g by A1; hence (G # x) . m <= g by MESFUNC5:def_13; ::_thesis: verum end; thus A9: ( F # x is convergent_to_finite_number implies G # x is convergent_to_finite_number ) ::_thesis: ( F # x is convergent implies G # x is convergent ) proof assume F # x is convergent_to_finite_number ; ::_thesis: G # x is convergent_to_finite_number then consider g being real number such that A10: lim (F # x) = g and A11: for p being real number st 0 < p holds ex n being Nat st for m being Nat st n <= m holds |.(((F # x) . m) - (lim (F # x))).| < p by Th7; for p being real number st 0 < p holds ex n being Nat st for m being Nat st n <= m holds |.(((G # x) . m) - (R_EAL g)).| < p proof let p be real number ; ::_thesis: ( 0 < p implies ex n being Nat st for m being Nat st n <= m holds |.(((G # x) . m) - (R_EAL g)).| < p ) assume 0 < p ; ::_thesis: ex n being Nat st for m being Nat st n <= m holds |.(((G # x) . m) - (R_EAL g)).| < p then consider n being Nat such that A12: for m being Nat st n <= m holds |.(((F # x) . m) - (lim (F # x))).| < p by A11; take n ; ::_thesis: for m being Nat st n <= m holds |.(((G # x) . m) - (R_EAL g)).| < p let m be Nat; ::_thesis: ( n <= m implies |.(((G # x) . m) - (R_EAL g)).| < p ) (F # x) . m = (F . m) . x by MESFUNC5:def_13; then (F # x) . m = ((F . m) | D) . x by A2, FUNCT_1:49; then A13: (F # x) . m = (G . m) . x by A1; assume n <= m ; ::_thesis: |.(((G # x) . m) - (R_EAL g)).| < p then |.(((F # x) . m) - (lim (F # x))).| < p by A12; hence |.(((G # x) . m) - (R_EAL g)).| < p by A10, A13, MESFUNC5:def_13; ::_thesis: verum end; hence G # x is convergent_to_finite_number by MESFUNC5:def_8; ::_thesis: verum end; assume A14: F # x is convergent ; ::_thesis: G # x is convergent percases ( F # x is convergent_to_+infty or F # x is convergent_to_-infty or F # x is convergent_to_finite_number ) by A14, MESFUNC5:def_11; suppose F # x is convergent_to_+infty ; ::_thesis: G # x is convergent hence G # x is convergent by A3, MESFUNC5:def_11; ::_thesis: verum end; suppose F # x is convergent_to_-infty ; ::_thesis: G # x is convergent hence G # x is convergent by A6, MESFUNC5:def_11; ::_thesis: verum end; suppose F # x is convergent_to_finite_number ; ::_thesis: G # x is convergent hence G # x is convergent by A9, MESFUNC5:def_11; ::_thesis: verum end; end; end; theorem Th13: :: MESFUNC9:13 for X being non empty set for S being SigmaField of X for M being sigma_Measure of S for E being Element of S for f being PartFunc of X,ExtREAL st E = dom f & f is_measurable_on E & f is nonnegative & M . (E /\ (eq_dom (f,+infty))) <> 0 holds Integral (M,f) = +infty proof let X be non empty set ; ::_thesis: for S being SigmaField of X for M being sigma_Measure of S for E being Element of S for f being PartFunc of X,ExtREAL st E = dom f & f is_measurable_on E & f is nonnegative & M . (E /\ (eq_dom (f,+infty))) <> 0 holds Integral (M,f) = +infty let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S for E being Element of S for f being PartFunc of X,ExtREAL st E = dom f & f is_measurable_on E & f is nonnegative & M . (E /\ (eq_dom (f,+infty))) <> 0 holds Integral (M,f) = +infty let M be sigma_Measure of S; ::_thesis: for E being Element of S for f being PartFunc of X,ExtREAL st E = dom f & f is_measurable_on E & f is nonnegative & M . (E /\ (eq_dom (f,+infty))) <> 0 holds Integral (M,f) = +infty let E be Element of S; ::_thesis: for f being PartFunc of X,ExtREAL st E = dom f & f is_measurable_on E & f is nonnegative & M . (E /\ (eq_dom (f,+infty))) <> 0 holds Integral (M,f) = +infty let f be PartFunc of X,ExtREAL; ::_thesis: ( E = dom f & f is_measurable_on E & f is nonnegative & M . (E /\ (eq_dom (f,+infty))) <> 0 implies Integral (M,f) = +infty ) assume that A1: E = dom f and A2: f is_measurable_on E and A3: f is nonnegative and A4: M . (E /\ (eq_dom (f,+infty))) <> 0 ; ::_thesis: Integral (M,f) = +infty reconsider EE = E /\ (eq_dom (f,+infty)) as Element of S by A1, A2, MESFUNC1:33; A5: dom (f | E) = E by A1, RELAT_1:62; E = (dom f) /\ E by A1; then A6: f | E is_measurable_on E by A2, MESFUNC5:42; integral+ (M,(f | EE)) <= integral+ (M,(f | E)) by A1, A2, A3, MESFUNC5:83, XBOOLE_1:17; then A7: integral+ (M,(f | EE)) <= Integral (M,(f | E)) by A3, A6, A5, MESFUNC5:15, MESFUNC5:88; A8: EE = (dom f) /\ EE by A1, XBOOLE_1:17, XBOOLE_1:28; f is_measurable_on EE by A2, MESFUNC1:30, XBOOLE_1:17; then A9: f | EE is_measurable_on EE by A8, MESFUNC5:42; A10: f | EE is nonnegative by A3, MESFUNC5:15; reconsider ES = {} as Element of S by PROB_1:4; deffunc H1( Element of NAT ) -> Element of bool [:X,ExtREAL:] = $1 (#) ((chi (EE,X)) | EE); consider G being Function such that A11: ( dom G = NAT & ( for n being Element of NAT holds G . n = H1(n) ) ) from FUNCT_1:sch_4(); now__::_thesis:_for_g_being_set_st_g_in_rng_G_holds_ g_in_PFuncs_(X,ExtREAL) let g be set ; ::_thesis: ( g in rng G implies g in PFuncs (X,ExtREAL) ) assume g in rng G ; ::_thesis: g in PFuncs (X,ExtREAL) then consider m being set such that A12: m in dom G and A13: g = G . m by FUNCT_1:def_3; reconsider m = m as Element of NAT by A11, A12; g = m (#) ((chi (EE,X)) | EE) by A11, A13; hence g in PFuncs (X,ExtREAL) by PARTFUN1:45; ::_thesis: verum end; then rng G c= PFuncs (X,ExtREAL) by TARSKI:def_3; then reconsider G = G as Functional_Sequence of X,ExtREAL by A11, FUNCT_2:def_1, RELSET_1:4; A14: for n being Nat holds ( dom (G . n) = EE & ( for x being set st x in dom (G . n) holds (G . n) . x = n ) ) proof let n be Nat; ::_thesis: ( dom (G . n) = EE & ( for x being set st x in dom (G . n) holds (G . n) . x = n ) ) reconsider n1 = n as Element of NAT by ORDINAL1:def_12; EE c= X ; then EE c= dom (chi (EE,X)) by FUNCT_3:def_3; then A15: dom ((chi (EE,X)) | EE) = EE by RELAT_1:62; A16: G . n = n1 (#) ((chi (EE,X)) | EE) by A11; hence A17: dom (G . n) = EE by A15, MESFUNC1:def_6; ::_thesis: for x being set st x in dom (G . n) holds (G . n) . x = n let x be set ; ::_thesis: ( x in dom (G . n) implies (G . n) . x = n ) assume A18: x in dom (G . n) ; ::_thesis: (G . n) . x = n then reconsider x1 = x as Element of X ; (chi (EE,X)) . x1 = 1. by A17, A18, FUNCT_3:def_3; then ((chi (EE,X)) | EE) . x1 = 1. by A15, A17, A18, FUNCT_1:47; then (G . n) . x = (R_EAL n1) * 1. by A16, A18, MESFUNC1:def_6; hence (G . n) . x = n by XXREAL_3:81; ::_thesis: verum end; A19: for n being Nat holds G . n is nonnegative proof let n be Nat; ::_thesis: G . n is nonnegative for x being set st x in dom (G . n) holds 0 <= (G . n) . x by A14; hence G . n is nonnegative by SUPINF_2:52; ::_thesis: verum end; deffunc H2( Element of NAT ) -> Element of ExtREAL = integral' (M,(G . $1)); consider K being Function of NAT,ExtREAL such that A20: for n being Element of NAT holds K . n = H2(n) from FUNCT_2:sch_4(); reconsider K = K as ExtREAL_sequence ; A21: for n being Nat holds K . n = integral' (M,(G . n)) proof let n be Nat; ::_thesis: K . n = integral' (M,(G . n)) reconsider n1 = n as Element of NAT by ORDINAL1:def_12; K . n = integral' (M,(G . n1)) by A20; hence K . n = integral' (M,(G . n)) ; ::_thesis: verum end; A22: dom (f | EE) = EE by A1, RELAT_1:62, XBOOLE_1:17; A23: for n, m being Nat st n <= m holds for x being Element of X st x in dom (f | EE) holds (G . n) . x <= (G . m) . x proof let n, m be Nat; ::_thesis: ( n <= m implies for x being Element of X st x in dom (f | EE) holds (G . n) . x <= (G . m) . x ) assume A24: n <= m ; ::_thesis: for x being Element of X st x in dom (f | EE) holds (G . n) . x <= (G . m) . x let x be Element of X; ::_thesis: ( x in dom (f | EE) implies (G . n) . x <= (G . m) . x ) assume A25: x in dom (f | EE) ; ::_thesis: (G . n) . x <= (G . m) . x then x in dom (G . n) by A22, A14; then A26: (G . n) . x = n by A14; x in dom (G . m) by A22, A14, A25; hence (G . n) . x <= (G . m) . x by A14, A24, A26; ::_thesis: verum end; A27: for n being Nat holds ( dom (G . n) = dom (f | EE) & G . n is_simple_func_in S ) proof let n be Nat; ::_thesis: ( dom (G . n) = dom (f | EE) & G . n is_simple_func_in S ) reconsider n1 = n as Element of NAT by ORDINAL1:def_12; thus A28: dom (G . n) = dom (f | EE) by A22, A14; ::_thesis: G . n is_simple_func_in S for x being set st x in dom (G . n) holds (G . n) . x = n1 by A14; hence G . n is_simple_func_in S by A22, A28, MESFUNC6:2; ::_thesis: verum end; A29: for i being Element of NAT holds K . i = (R_EAL i) * (M . (dom (G . i))) proof let i be Element of NAT ; ::_thesis: K . i = (R_EAL i) * (M . (dom (G . i))) for x being set st x in dom (G . i) holds (G . i) . x = R_EAL i by A14; then integral' (M,(G . i)) = (R_EAL i) * (M . (dom (G . i))) by A27, MESFUNC5:71; hence K . i = (R_EAL i) * (M . (dom (G . i))) by A21; ::_thesis: verum end; M . ES <= M . EE by MEASURE1:31, XBOOLE_1:2; then A30: 0 < M . EE by A4, VALUED_0:def_19; A31: not rng K is bounded_above proof assume rng K is bounded_above ; ::_thesis: contradiction then consider UB being real number such that A32: UB is UpperBound of rng K by XXREAL_2:def_10; UB in REAL by XREAL_0:def_1; then reconsider r = UB as Real ; percases ( M . EE = +infty or M . EE in REAL ) by A30, XXREAL_0:10; supposeA33: M . EE = +infty ; ::_thesis: contradiction K . 1 = (R_EAL 1) * (M . (dom (G . 1))) by A29; then K . 1 = (R_EAL 1) * (M . EE) by A14; then A34: K . 1 = +infty by A33, XXREAL_3:def_5; dom K = NAT by FUNCT_2:def_1; then K . 1 in rng K by FUNCT_1:3; then K . 1 <= UB by A32, XXREAL_2:def_1; hence contradiction by A34, XXREAL_0:4; ::_thesis: verum end; suppose M . EE in REAL ; ::_thesis: contradiction then reconsider ee = M . EE as Real ; consider n being Element of NAT such that A35: r / ee < n by SEQ_4:3; K . n = (R_EAL n) * (M . (dom (G . n))) by A29; then K . n = (R_EAL n) * (M . EE) by A14; then A36: K . n = n * ee by EXTREAL1:1; (r / ee) * ee < n * ee by A30, A35, XREAL_1:68; then r / (ee / ee) < K . n by A36, XCMPLX_1:82; then A37: r < K . n by A4, XCMPLX_1:51; dom K = NAT by FUNCT_2:def_1; then K . n in rng K by FUNCT_1:3; then K . n <= r by A32, XXREAL_2:def_1; hence contradiction by A37; ::_thesis: verum end; end; end; for n, m being Element of NAT st m <= n holds K . m <= K . n proof let n, m be Element of NAT ; ::_thesis: ( m <= n implies K . m <= K . n ) dom (G . m) = EE by A14; then A38: K . m = (R_EAL m) * (M . EE) by A29; dom (G . n) = EE by A14; then A39: K . n = (R_EAL n) * (M . EE) by A29; assume m <= n ; ::_thesis: K . m <= K . n hence K . m <= K . n by A30, A38, A39, XXREAL_3:71; ::_thesis: verum end; then A40: K is non-decreasing by RINFSUP2:7; then A41: lim K = sup K by RINFSUP2:37; A42: for x being Element of X st x in dom (f | EE) holds ( G # x is convergent & lim (G # x) = (f | EE) . x ) proof let x be Element of X; ::_thesis: ( x in dom (f | EE) implies ( G # x is convergent & lim (G # x) = (f | EE) . x ) ) assume A43: x in dom (f | EE) ; ::_thesis: ( G # x is convergent & lim (G # x) = (f | EE) . x ) then A44: x in EE by A1, RELAT_1:62, XBOOLE_1:17; then x in eq_dom (f,+infty) by XBOOLE_0:def_4; then f . x = +infty by MESFUNC1:def_15; then A45: (f | EE) . x = +infty by A44, FUNCT_1:49; A46: not rng (G # x) is bounded_above proof assume rng (G # x) is bounded_above ; ::_thesis: contradiction then consider UB being real number such that A47: UB is UpperBound of rng (G # x) by XXREAL_2:def_10; UB in REAL by XREAL_0:def_1; then reconsider r = UB as Real ; consider n being Element of NAT such that A48: r < n by SEQ_4:3; x in dom (G . n) by A14, A44; then (G . n) . x = n by A14; then A49: UB < (G # x) . n by A48, MESFUNC5:def_13; dom (G # x) = NAT by FUNCT_2:def_1; then (G # x) . n in rng (G # x) by FUNCT_1:3; hence contradiction by A49, A47, XXREAL_2:def_1; ::_thesis: verum end; for n, m being Element of NAT st m <= n holds (G # x) . m <= (G # x) . n proof let n, m be Element of NAT ; ::_thesis: ( m <= n implies (G # x) . m <= (G # x) . n ) dom (G . n) = EE by A14; then A50: (G . n) . x = n by A22, A14, A43; dom (G . m) = EE by A14; then (G . m) . x = m by A22, A14, A43; then A51: (G # x) . m = m by MESFUNC5:def_13; assume m <= n ; ::_thesis: (G # x) . m <= (G # x) . n hence (G # x) . m <= (G # x) . n by A50, A51, MESFUNC5:def_13; ::_thesis: verum end; then A52: G # x is non-decreasing by RINFSUP2:7; sup (rng (G # x)) is UpperBound of rng (G # x) by XXREAL_2:def_3; then sup (G # x) = +infty by A46, XXREAL_2:53; hence ( G # x is convergent & lim (G # x) = (f | EE) . x ) by A52, A45, RINFSUP2:37; ::_thesis: verum end; sup (rng K) is UpperBound of rng K by XXREAL_2:def_3; then A53: sup K = +infty by A31, XXREAL_2:53; K is convergent by A40, RINFSUP2:37; then integral+ (M,(f | EE)) = +infty by A10, A22, A9, A27, A19, A23, A42, A21, A41, A53, MESFUNC5:def_15; then Integral (M,(f | E)) = +infty by A7, XXREAL_0:4; hence Integral (M,f) = +infty by A1, RELAT_1:68; ::_thesis: verum end; theorem :: MESFUNC9:14 for X being non empty set for S being SigmaField of X for M being sigma_Measure of S for E being Element of S holds ( Integral (M,(chi (E,X))) = M . E & Integral (M,((chi (E,X)) | E)) = M . E ) proof let X be non empty set ; ::_thesis: for S being SigmaField of X for M being sigma_Measure of S for E being Element of S holds ( Integral (M,(chi (E,X))) = M . E & Integral (M,((chi (E,X)) | E)) = M . E ) let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S for E being Element of S holds ( Integral (M,(chi (E,X))) = M . E & Integral (M,((chi (E,X)) | E)) = M . E ) let M be sigma_Measure of S; ::_thesis: for E being Element of S holds ( Integral (M,(chi (E,X))) = M . E & Integral (M,((chi (E,X)) | E)) = M . E ) let E be Element of S; ::_thesis: ( Integral (M,(chi (E,X))) = M . E & Integral (M,((chi (E,X)) | E)) = M . E ) reconsider XX = X as Element of S by MEASURE1:7; set F = XX \ E; A1: now__::_thesis:_for_x_being_Element_of_X_st_x_in_dom_(max-_(chi_(E,X)))_holds_ (max-_(chi_(E,X)))_._x_=_0 let x be Element of X; ::_thesis: ( x in dom (max- (chi (E,X))) implies (max- (chi (E,X))) . b1 = 0 ) assume A2: x in dom (max- (chi (E,X))) ; ::_thesis: (max- (chi (E,X))) . b1 = 0 percases ( x in E or not x in E ) ; suppose x in E ; ::_thesis: (max- (chi (E,X))) . b1 = 0 then (chi (E,X)) . x = 1 by FUNCT_3:def_3; then max ((- ((chi (E,X)) . x)),0.) = 0. by XXREAL_0:def_10; hence (max- (chi (E,X))) . x = 0 by A2, MESFUNC2:def_3; ::_thesis: verum end; suppose not x in E ; ::_thesis: (max- (chi (E,X))) . b1 = 0 then (chi (E,X)) . x = 0. by FUNCT_3:def_3; then - ((chi (E,X)) . x) = 0 ; then max ((- ((chi (E,X)) . x)),0.) = 0 ; hence (max- (chi (E,X))) . x = 0 by A2, MESFUNC2:def_3; ::_thesis: verum end; end; end; A3: XX = dom (chi (E,X)) by FUNCT_3:def_3; then A4: XX = dom (max+ (chi (E,X))) by MESFUNC7:23; A5: XX /\ (XX \ E) = XX \ E by XBOOLE_1:28; then A6: dom ((max+ (chi (E,X))) | (XX \ E)) = XX \ E by A4, RELAT_1:61; A7: now__::_thesis:_for_x_being_Element_of_X_st_x_in_dom_((max+_(chi_(E,X)))_|_(XX_\_E))_holds_ ((max+_(chi_(E,X)))_|_(XX_\_E))_._x_=_0 let x be Element of X; ::_thesis: ( x in dom ((max+ (chi (E,X))) | (XX \ E)) implies ((max+ (chi (E,X))) | (XX \ E)) . x = 0 ) assume A8: x in dom ((max+ (chi (E,X))) | (XX \ E)) ; ::_thesis: ((max+ (chi (E,X))) | (XX \ E)) . x = 0 then (chi (E,X)) . x = 0 by A6, FUNCT_3:37; then (max+ (chi (E,X))) . x = 0 by MESFUNC7:23; hence ((max+ (chi (E,X))) | (XX \ E)) . x = 0 by A8, FUNCT_1:47; ::_thesis: verum end; A9: chi (E,X) is_measurable_on XX by MESFUNC2:29; then A10: max+ (chi (E,X)) is_measurable_on XX by MESFUNC7:23; then max+ (chi (E,X)) is_measurable_on XX \ E by MESFUNC1:30; then A11: integral+ (M,((max+ (chi (E,X))) | (XX \ E))) = 0 by A4, A5, A6, A7, MESFUNC5:42, MESFUNC5:87; A12: XX /\ E = E by XBOOLE_1:28; then A13: dom ((max+ (chi (E,X))) | E) = E by A4, RELAT_1:61; E \/ (XX \ E) = XX by A12, XBOOLE_1:51; then A14: (max+ (chi (E,X))) | (E \/ (XX \ E)) = max+ (chi (E,X)) by A4, RELAT_1:69; A15: for x being set st x in dom (max+ (chi (E,X))) holds 0. <= (max+ (chi (E,X))) . x by MESFUNC2:12; then A16: max+ (chi (E,X)) is nonnegative by SUPINF_2:52; then integral+ (M,((max+ (chi (E,X))) | (E \/ (XX \ E)))) = (integral+ (M,((max+ (chi (E,X))) | E))) + (integral+ (M,((max+ (chi (E,X))) | (XX \ E)))) by A4, A10, MESFUNC5:81, XBOOLE_1:79; then A17: integral+ (M,(max+ (chi (E,X)))) = integral+ (M,((max+ (chi (E,X))) | E)) by A14, A11, XXREAL_3:4; A18: now__::_thesis:_for_x_being_set_st_x_in_dom_((max+_(chi_(E,X)))_|_E)_holds_ ((max+_(chi_(E,X)))_|_E)_._x_=_1 let x be set ; ::_thesis: ( x in dom ((max+ (chi (E,X))) | E) implies ((max+ (chi (E,X))) | E) . x = 1 ) assume A19: x in dom ((max+ (chi (E,X))) | E) ; ::_thesis: ((max+ (chi (E,X))) | E) . x = 1 then (chi (E,X)) . x = 1 by A13, FUNCT_3:def_3; then (max+ (chi (E,X))) . x = 1 by MESFUNC7:23; hence ((max+ (chi (E,X))) | E) . x = 1 by A19, FUNCT_1:47; ::_thesis: verum end; then (max+ (chi (E,X))) | E is_simple_func_in S by A13, MESFUNC6:2; then integral+ (M,(max+ (chi (E,X)))) = integral' (M,((max+ (chi (E,X))) | E)) by A16, A17, MESFUNC5:15, MESFUNC5:77; then A20: integral+ (M,(max+ (chi (E,X)))) = (R_EAL 1) * (M . (dom ((max+ (chi (E,X))) | E))) by A13, A18, MESFUNC5:104; max+ (chi (E,X)) is_measurable_on E by A10, MESFUNC1:30; then (max+ (chi (E,X))) | E is_measurable_on E by A4, A12, MESFUNC5:42; then A21: (chi (E,X)) | E is_measurable_on E by MESFUNC7:23; (max+ (chi (E,X))) | E is nonnegative by A15, MESFUNC5:15, SUPINF_2:52; then A22: (chi (E,X)) | E is nonnegative by MESFUNC7:23; E = dom ((chi (E,X)) | E) by A13, MESFUNC7:23; then A23: Integral (M,((chi (E,X)) | E)) = integral+ (M,((chi (E,X)) | E)) by A21, A22, MESFUNC5:88; XX = dom (max- (chi (E,X))) by A3, MESFUNC2:def_3; then integral+ (M,(max- (chi (E,X)))) = 0 by A3, A9, A1, MESFUNC2:26, MESFUNC5:87; then Integral (M,(chi (E,X))) = (R_EAL 1) * (M . E) by A13, A20, XXREAL_3:15; hence Integral (M,(chi (E,X))) = M . E by XXREAL_3:81; ::_thesis: Integral (M,((chi (E,X)) | E)) = M . E (chi (E,X)) | E = (max+ (chi (E,X))) | E by MESFUNC7:23; hence Integral (M,((chi (E,X)) | E)) = M . E by A13, A17, A20, A23, XXREAL_3:81; ::_thesis: verum end; theorem Th15: :: MESFUNC9:15 for X being non empty set for S being SigmaField of X for M being sigma_Measure of S for E being Element of S for f, g being PartFunc of X,ExtREAL st E c= dom f & E c= dom g & f is_measurable_on E & g is_measurable_on E & f is nonnegative & ( for x being Element of X st x in E holds f . x <= g . x ) holds Integral (M,(f | E)) <= Integral (M,(g | E)) proof let X be non empty set ; ::_thesis: for S being SigmaField of X for M being sigma_Measure of S for E being Element of S for f, g being PartFunc of X,ExtREAL st E c= dom f & E c= dom g & f is_measurable_on E & g is_measurable_on E & f is nonnegative & ( for x being Element of X st x in E holds f . x <= g . x ) holds Integral (M,(f | E)) <= Integral (M,(g | E)) let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S for E being Element of S for f, g being PartFunc of X,ExtREAL st E c= dom f & E c= dom g & f is_measurable_on E & g is_measurable_on E & f is nonnegative & ( for x being Element of X st x in E holds f . x <= g . x ) holds Integral (M,(f | E)) <= Integral (M,(g | E)) let M be sigma_Measure of S; ::_thesis: for E being Element of S for f, g being PartFunc of X,ExtREAL st E c= dom f & E c= dom g & f is_measurable_on E & g is_measurable_on E & f is nonnegative & ( for x being Element of X st x in E holds f . x <= g . x ) holds Integral (M,(f | E)) <= Integral (M,(g | E)) let E be Element of S; ::_thesis: for f, g being PartFunc of X,ExtREAL st E c= dom f & E c= dom g & f is_measurable_on E & g is_measurable_on E & f is nonnegative & ( for x being Element of X st x in E holds f . x <= g . x ) holds Integral (M,(f | E)) <= Integral (M,(g | E)) let f, g be PartFunc of X,ExtREAL; ::_thesis: ( E c= dom f & E c= dom g & f is_measurable_on E & g is_measurable_on E & f is nonnegative & ( for x being Element of X st x in E holds f . x <= g . x ) implies Integral (M,(f | E)) <= Integral (M,(g | E)) ) assume that A1: E c= dom f and A2: E c= dom g and A3: f is_measurable_on E and A4: g is_measurable_on E and A5: f is nonnegative and A6: for x being Element of X st x in E holds f . x <= g . x ; ::_thesis: Integral (M,(f | E)) <= Integral (M,(g | E)) set F2 = g | E; A7: E = dom (f | E) by A1, RELAT_1:62; set F1 = f | E; A8: f | E is nonnegative by A5, MESFUNC5:15; A9: E = dom (g | E) by A2, RELAT_1:62; A10: for x being Element of X st x in dom (f | E) holds (f | E) . x <= (g | E) . x proof let x be Element of X; ::_thesis: ( x in dom (f | E) implies (f | E) . x <= (g | E) . x ) assume A11: x in dom (f | E) ; ::_thesis: (f | E) . x <= (g | E) . x then A12: (f | E) . x = f . x by FUNCT_1:47; (g | E) . x = g . x by A7, A9, A11, FUNCT_1:47; hence (f | E) . x <= (g | E) . x by A6, A7, A11, A12; ::_thesis: verum end; for x being set st x in dom (g | E) holds 0 <= (g | E) . x proof let x be set ; ::_thesis: ( x in dom (g | E) implies 0 <= (g | E) . x ) assume A13: x in dom (g | E) ; ::_thesis: 0 <= (g | E) . x 0 <= (f | E) . x by A8, SUPINF_2:51; hence 0 <= (g | E) . x by A7, A9, A10, A13; ::_thesis: verum end; then A14: g | E is nonnegative by SUPINF_2:52; A15: (dom g) /\ E = E by A2, XBOOLE_1:28; then A16: g | E is_measurable_on E by A4, MESFUNC5:42; A17: (dom f) /\ E = E by A1, XBOOLE_1:28; then f | E is_measurable_on E by A3, MESFUNC5:42; then integral+ (M,(f | E)) <= integral+ (M,(g | E)) by A8, A7, A9, A10, A14, A16, MESFUNC5:85; then Integral (M,(f | E)) <= integral+ (M,(g | E)) by A3, A8, A7, A17, MESFUNC5:42, MESFUNC5:88; hence Integral (M,(f | E)) <= Integral (M,(g | E)) by A4, A9, A14, A15, MESFUNC5:42, MESFUNC5:88; ::_thesis: verum end; begin definition let f be ext-real-valued Function; let x be set ; :: original: . redefine funcf . x -> Element of ExtREAL ; coherence f . x is Element of ExtREAL by XXREAL_0:def_1; end; definition let s be ext-real-valued Function; func Partial_Sums s -> ExtREAL_sequence means :Def1: :: MESFUNC9:def 1 ( it . 0 = s . 0 & ( for n being Nat holds it . (n + 1) = (it . n) + (s . (n + 1)) ) ); existence ex b1 being ExtREAL_sequence st ( b1 . 0 = s . 0 & ( for n being Nat holds b1 . (n + 1) = (b1 . n) + (s . (n + 1)) ) ) proof deffunc H1( Nat, R_eal) -> Element of ExtREAL = $2 + (s . ($1 + 1)); consider f being Function of NAT,ExtREAL such that A1: ( f . 0 = s . 0 & ( for n being Nat holds f . (n + 1) = H1(n,f . n) ) ) from NAT_1:sch_12(); take f ; ::_thesis: ( f . 0 = s . 0 & ( for n being Nat holds f . (n + 1) = (f . n) + (s . (n + 1)) ) ) thus ( f . 0 = s . 0 & ( for n being Nat holds f . (n + 1) = (f . n) + (s . (n + 1)) ) ) by A1; ::_thesis: verum end; uniqueness for b1, b2 being ExtREAL_sequence st b1 . 0 = s . 0 & ( for n being Nat holds b1 . (n + 1) = (b1 . n) + (s . (n + 1)) ) & b2 . 0 = s . 0 & ( for n being Nat holds b2 . (n + 1) = (b2 . n) + (s . (n + 1)) ) holds b1 = b2 proof let s1, s2 be ExtREAL_sequence; ::_thesis: ( s1 . 0 = s . 0 & ( for n being Nat holds s1 . (n + 1) = (s1 . n) + (s . (n + 1)) ) & s2 . 0 = s . 0 & ( for n being Nat holds s2 . (n + 1) = (s2 . n) + (s . (n + 1)) ) implies s1 = s2 ) assume that A2: s1 . 0 = s . 0 and A3: for n being Nat holds s1 . (n + 1) = (s1 . n) + (s . (n + 1)) and A4: s2 . 0 = s . 0 and A5: for n being Nat holds s2 . (n + 1) = (s2 . n) + (s . (n + 1)) ; ::_thesis: s1 = s2 defpred S1[ Element of NAT ] means s1 . $1 = s2 . $1; A6: for k being Element of NAT st S1[k] holds S1[k + 1] proof let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] ) assume s1 . k = s2 . k ; ::_thesis: S1[k + 1] hence s1 . (k + 1) = (s2 . k) + (s . (k + 1)) by A3 .= s2 . (k + 1) by A5 ; ::_thesis: verum end; A7: S1[ 0 ] by A2, A4; for n being Element of NAT holds S1[n] from NAT_1:sch_1(A7, A6); hence s1 = s2 by FUNCT_2:63; ::_thesis: verum end; end; :: deftheorem Def1 defines Partial_Sums MESFUNC9:def_1_:_ for s being ext-real-valued Function for b2 being ExtREAL_sequence holds ( b2 = Partial_Sums s iff ( b2 . 0 = s . 0 & ( for n being Nat holds b2 . (n + 1) = (b2 . n) + (s . (n + 1)) ) ) ); definition let s be ext-real-valued Function; attrs is summable means :Def2: :: MESFUNC9:def 2 Partial_Sums s is convergent ; end; :: deftheorem Def2 defines summable MESFUNC9:def_2_:_ for s being ext-real-valued Function holds ( s is summable iff Partial_Sums s is convergent ); definition let s be ext-real-valued Function; func Sum s -> R_eal equals :: MESFUNC9:def 3 lim (Partial_Sums s); correctness coherence lim (Partial_Sums s) is R_eal; ; end; :: deftheorem defines Sum MESFUNC9:def_3_:_ for s being ext-real-valued Function holds Sum s = lim (Partial_Sums s); theorem Th16: :: MESFUNC9:16 for seq being ExtREAL_sequence st seq is V111() holds ( Partial_Sums seq is V111() & Partial_Sums seq is non-decreasing ) proof let seq be ExtREAL_sequence; ::_thesis: ( seq is V111() implies ( Partial_Sums seq is V111() & Partial_Sums seq is non-decreasing ) ) set PS = Partial_Sums seq; defpred S1[ Nat] means 0 <= (Partial_Sums seq) . $1; assume A1: seq is V111() ; ::_thesis: ( Partial_Sums seq is V111() & Partial_Sums seq is non-decreasing ) A2: for k being Nat st S1[k] holds S1[k + 1] proof let k be Nat; ::_thesis: ( S1[k] implies S1[k + 1] ) assume A3: S1[k] ; ::_thesis: S1[k + 1] A4: (Partial_Sums seq) . (k + 1) = ((Partial_Sums seq) . k) + (seq . (k + 1)) by Def1; seq . (k + 1) >= 0 by A1, SUPINF_2:51; hence S1[k + 1] by A3, A4; ::_thesis: verum end; (Partial_Sums seq) . 0 = seq . 0 by Def1; then A5: S1[ 0 ] by A1, SUPINF_2:51; for m being Nat holds S1[m] from NAT_1:sch_2(A5, A2); then for k being set st k in dom (Partial_Sums seq) holds 0 <= (Partial_Sums seq) . k ; hence Partial_Sums seq is V111() by SUPINF_2:52; ::_thesis: Partial_Sums seq is non-decreasing for n, m being Element of NAT st m <= n holds (Partial_Sums seq) . m <= (Partial_Sums seq) . n proof let n, m be Element of NAT ; ::_thesis: ( m <= n implies (Partial_Sums seq) . m <= (Partial_Sums seq) . n ) reconsider m1 = m as Nat ; defpred S2[ Nat] means (Partial_Sums seq) . m1 <= (Partial_Sums seq) . $1; A6: for k being Nat holds (Partial_Sums seq) . k <= (Partial_Sums seq) . (k + 1) proof let k be Nat; ::_thesis: (Partial_Sums seq) . k <= (Partial_Sums seq) . (k + 1) A7: 0 <= seq . (k + 1) by A1, SUPINF_2:51; (Partial_Sums seq) . (k + 1) = ((Partial_Sums seq) . k) + (seq . (k + 1)) by Def1; hence (Partial_Sums seq) . k <= (Partial_Sums seq) . (k + 1) by A7, XXREAL_3:39; ::_thesis: verum end; A8: for k being Nat st k >= m1 & ( for l being Nat st l >= m1 & l < k holds S2[l] ) holds S2[k] proof let k be Nat; ::_thesis: ( k >= m1 & ( for l being Nat st l >= m1 & l < k holds S2[l] ) implies S2[k] ) assume that A9: k >= m1 and A10: for l being Nat st l >= m1 & l < k holds S2[l] ; ::_thesis: S2[k] now__::_thesis:_(_k_>_m1_implies_S2[k]_) assume k > m1 ; ::_thesis: S2[k] then A11: k >= m1 + 1 by NAT_1:13; percases ( k = m1 + 1 or k > m1 + 1 ) by A11, XXREAL_0:1; suppose k = m1 + 1 ; ::_thesis: S2[k] hence S2[k] by A6; ::_thesis: verum end; supposeA12: k > m1 + 1 ; ::_thesis: S2[k] then reconsider l = k - 1 as Element of NAT by NAT_1:20; k < k + 1 by NAT_1:13; then A13: k > l by XREAL_1:19; k = l + 1 ; then A14: (Partial_Sums seq) . l <= (Partial_Sums seq) . k by A6; l >= m1 by A12, XREAL_1:19; then (Partial_Sums seq) . m1 <= (Partial_Sums seq) . l by A10, A13; hence S2[k] by A14, XXREAL_0:2; ::_thesis: verum end; end; end; hence S2[k] by A9, XXREAL_0:1; ::_thesis: verum end; A15: for k being Nat st k >= m1 holds S2[k] from NAT_1:sch_9(A8); assume m <= n ; ::_thesis: (Partial_Sums seq) . m <= (Partial_Sums seq) . n hence (Partial_Sums seq) . m <= (Partial_Sums seq) . n by A15; ::_thesis: verum end; hence Partial_Sums seq is non-decreasing by RINFSUP2:7; ::_thesis: verum end; theorem :: MESFUNC9:17 for seq being ExtREAL_sequence st ( for n being Nat holds 0 < seq . n ) holds for m being Nat holds 0 < (Partial_Sums seq) . m proof let seq be ExtREAL_sequence; ::_thesis: ( ( for n being Nat holds 0 < seq . n ) implies for m being Nat holds 0 < (Partial_Sums seq) . m ) defpred S1[ Nat] means 0 < (Partial_Sums seq) . $1; assume A1: for n being Nat holds 0 < seq . n ; ::_thesis: for m being Nat holds 0 < (Partial_Sums seq) . m A2: for k being Nat st S1[k] holds S1[k + 1] proof let k be Nat; ::_thesis: ( S1[k] implies S1[k + 1] ) assume A3: S1[k] ; ::_thesis: S1[k + 1] A4: (Partial_Sums seq) . (k + 1) = ((Partial_Sums seq) . k) + (seq . (k + 1)) by Def1; seq . (k + 1) > 0 by A1; hence S1[k + 1] by A3, A4; ::_thesis: verum end; (Partial_Sums seq) . 0 = seq . 0 by Def1; then A5: S1[ 0 ] by A1; thus for m being Nat holds S1[m] from NAT_1:sch_2(A5, A2); ::_thesis: verum end; theorem Th18: :: MESFUNC9:18 for X being non empty set for F, G being Functional_Sequence of X,ExtREAL for D being set st F is with_the_same_dom & ( for n being Nat holds G . n = (F . n) | D ) holds G is with_the_same_dom proof let X be non empty set ; ::_thesis: for F, G being Functional_Sequence of X,ExtREAL for D being set st F is with_the_same_dom & ( for n being Nat holds G . n = (F . n) | D ) holds G is with_the_same_dom let F, G be Functional_Sequence of X,ExtREAL; ::_thesis: for D being set st F is with_the_same_dom & ( for n being Nat holds G . n = (F . n) | D ) holds G is with_the_same_dom let D be set ; ::_thesis: ( F is with_the_same_dom & ( for n being Nat holds G . n = (F . n) | D ) implies G is with_the_same_dom ) assume that A1: F is with_the_same_dom and A2: for n being Nat holds G . n = (F . n) | D ; ::_thesis: G is with_the_same_dom let n, m be Nat; :: according to MESFUNC8:def_2 ::_thesis: dom (G . n) = dom (G . m) G . m = (F . m) | D by A2; then A3: dom (G . m) = (dom (F . m)) /\ D by RELAT_1:61; G . n = (F . n) | D by A2; then dom (G . n) = (dom (F . n)) /\ D by RELAT_1:61; hence dom (G . n) = dom (G . m) by A1, A3, MESFUNC8:def_2; ::_thesis: verum end; theorem Th19: :: MESFUNC9:19 for X being non empty set for F, G being Functional_Sequence of X,ExtREAL for D being set st D c= dom (F . 0) & ( for n being Nat holds G . n = (F . n) | D ) & ( for x being Element of X st x in D holds F # x is convergent ) holds (lim F) | D = lim G proof let X be non empty set ; ::_thesis: for F, G being Functional_Sequence of X,ExtREAL for D being set st D c= dom (F . 0) & ( for n being Nat holds G . n = (F . n) | D ) & ( for x being Element of X st x in D holds F # x is convergent ) holds (lim F) | D = lim G let F, G be Functional_Sequence of X,ExtREAL; ::_thesis: for D being set st D c= dom (F . 0) & ( for n being Nat holds G . n = (F . n) | D ) & ( for x being Element of X st x in D holds F # x is convergent ) holds (lim F) | D = lim G let D be set ; ::_thesis: ( D c= dom (F . 0) & ( for n being Nat holds G . n = (F . n) | D ) & ( for x being Element of X st x in D holds F # x is convergent ) implies (lim F) | D = lim G ) assume that A1: D c= dom (F . 0) and A2: for n being Nat holds G . n = (F . n) | D and A3: for x being Element of X st x in D holds F # x is convergent ; ::_thesis: (lim F) | D = lim G G . 0 = (F . 0) | D by A2; then A4: dom (G . 0) = D by A1, RELAT_1:62; A5: dom ((lim F) | D) = (dom (lim F)) /\ D by RELAT_1:61; then dom ((lim F) | D) = (dom (F . 0)) /\ D by MESFUNC8:def_9; then dom ((lim F) | D) = D by A1, XBOOLE_1:28; then A6: dom ((lim F) | D) = dom (lim G) by A4, MESFUNC8:def_9; now__::_thesis:_for_x_being_Element_of_X_st_x_in_dom_((lim_F)_|_D)_holds_ (lim_G)_._x_=_((lim_F)_|_D)_._x let x be Element of X; ::_thesis: ( x in dom ((lim F) | D) implies (lim G) . b1 = ((lim F) | D) . b1 ) assume A7: x in dom ((lim F) | D) ; ::_thesis: (lim G) . b1 = ((lim F) | D) . b1 then A8: ((lim F) | D) . x = (lim F) . x by FUNCT_1:47; x in dom (lim F) by A5, A7, XBOOLE_0:def_4; then A9: ((lim F) | D) . x = lim (F # x) by A8, MESFUNC8:def_9; A10: x in D by A7, RELAT_1:57; then A11: F # x is convergent by A3; percases ( F # x is convergent_to_+infty or F # x is convergent_to_-infty or F # x is convergent_to_finite_number ) by A11, MESFUNC5:def_11; supposeA12: F # x is convergent_to_+infty ; ::_thesis: (lim G) . b1 = ((lim F) | D) . b1 then G # x is convergent_to_+infty by A2, A10, Th12; then lim (G # x) = +infty by Th7; then (lim G) . x = +infty by A6, A7, MESFUNC8:def_9; hence (lim G) . x = ((lim F) | D) . x by A9, A12, Th7; ::_thesis: verum end; supposeA13: F # x is convergent_to_-infty ; ::_thesis: (lim G) . b1 = ((lim F) | D) . b1 then G # x is convergent_to_-infty by A2, A10, Th12; then lim (G # x) = -infty by Th7; then (lim G) . x = -infty by A6, A7, MESFUNC8:def_9; hence (lim G) . x = ((lim F) | D) . x by A9, A13, Th7; ::_thesis: verum end; supposeA14: F # x is convergent_to_finite_number ; ::_thesis: (lim G) . b1 = ((lim F) | D) . b1 then consider g being real number such that A15: lim (F # x) = g and A16: for p being real number st 0 < p holds ex n being Nat st for m being Nat st n <= m holds |.(((F # x) . m) - (lim (F # x))).| < p by Th7; A17: now__::_thesis:_for_p_being_real_number_st_0_<_p_holds_ ex_n_being_Nat_st_ for_m_being_Nat_st_n_<=_m_holds_ |.(((G_#_x)_._m)_-_(R_EAL_g)).|_<_p let p be real number ; ::_thesis: ( 0 < p implies ex n being Nat st for m being Nat st n <= m holds |.(((G # x) . m) - (R_EAL g)).| < p ) assume 0 < p ; ::_thesis: ex n being Nat st for m being Nat st n <= m holds |.(((G # x) . m) - (R_EAL g)).| < p then consider n being Nat such that A18: for m being Nat st n <= m holds |.(((F # x) . m) - (lim (F # x))).| < p by A16; take n = n; ::_thesis: for m being Nat st n <= m holds |.(((G # x) . m) - (R_EAL g)).| < p let m be Nat; ::_thesis: ( n <= m implies |.(((G # x) . m) - (R_EAL g)).| < p ) (F # x) . m = (F . m) . x by MESFUNC5:def_13; then (F # x) . m = ((F . m) | D) . x by A10, FUNCT_1:49; then A19: (F # x) . m = (G . m) . x by A2; assume n <= m ; ::_thesis: |.(((G # x) . m) - (R_EAL g)).| < p then |.(((F # x) . m) - (lim (F # x))).| < p by A18; hence |.(((G # x) . m) - (R_EAL g)).| < p by A15, A19, MESFUNC5:def_13; ::_thesis: verum end; A20: G # x is convergent_to_finite_number by A2, A10, A14, Th12; then G # x is convergent by MESFUNC5:def_11; then lim (G # x) = R_EAL g by A17, A20, MESFUNC5:def_12; hence (lim G) . x = ((lim F) | D) . x by A6, A7, A9, A15, MESFUNC8:def_9; ::_thesis: verum end; end; end; hence (lim F) | D = lim G by A6, PARTFUN1:5; ::_thesis: verum end; theorem Th20: :: MESFUNC9:20 for X being non empty set for S being SigmaField of X for E being Element of S for F, G being Functional_Sequence of X,ExtREAL for n being Nat st F is with_the_same_dom & E c= dom (F . 0) & ( for m being Nat holds ( F . m is_measurable_on E & G . m = (F . m) | E ) ) holds G . n is_measurable_on E proof let X be non empty set ; ::_thesis: for S being SigmaField of X for E being Element of S for F, G being Functional_Sequence of X,ExtREAL for n being Nat st F is with_the_same_dom & E c= dom (F . 0) & ( for m being Nat holds ( F . m is_measurable_on E & G . m = (F . m) | E ) ) holds G . n is_measurable_on E let S be SigmaField of X; ::_thesis: for E being Element of S for F, G being Functional_Sequence of X,ExtREAL for n being Nat st F is with_the_same_dom & E c= dom (F . 0) & ( for m being Nat holds ( F . m is_measurable_on E & G . m = (F . m) | E ) ) holds G . n is_measurable_on E let E be Element of S; ::_thesis: for F, G being Functional_Sequence of X,ExtREAL for n being Nat st F is with_the_same_dom & E c= dom (F . 0) & ( for m being Nat holds ( F . m is_measurable_on E & G . m = (F . m) | E ) ) holds G . n is_measurable_on E let F, G be Functional_Sequence of X,ExtREAL; ::_thesis: for n being Nat st F is with_the_same_dom & E c= dom (F . 0) & ( for m being Nat holds ( F . m is_measurable_on E & G . m = (F . m) | E ) ) holds G . n is_measurable_on E let n be Nat; ::_thesis: ( F is with_the_same_dom & E c= dom (F . 0) & ( for m being Nat holds ( F . m is_measurable_on E & G . m = (F . m) | E ) ) implies G . n is_measurable_on E ) assume that A1: F is with_the_same_dom and A2: E c= dom (F . 0) and A3: for m being Nat holds ( F . m is_measurable_on E & G . m = (F . m) | E ) ; ::_thesis: G . n is_measurable_on E dom (F . n) = dom (F . 0) by A1, MESFUNC8:def_2; then (dom (F . n)) /\ E = E by A2, XBOOLE_1:28; then (F . n) | E is_measurable_on E by A3, MESFUNC5:42; hence G . n is_measurable_on E by A3; ::_thesis: verum end; theorem Th21: :: MESFUNC9:21 for X being non empty set for S being SigmaField of X for E being Element of S for F, G being Functional_Sequence of X,ExtREAL st E c= dom (F . 0) & G is with_the_same_dom & ( for x being Element of X st x in E holds F # x is summable ) & ( for n being Nat holds G . n = (F . n) | E ) holds for x being Element of X st x in E holds G # x is summable proof let X be non empty set ; ::_thesis: for S being SigmaField of X for E being Element of S for F, G being Functional_Sequence of X,ExtREAL st E c= dom (F . 0) & G is with_the_same_dom & ( for x being Element of X st x in E holds F # x is summable ) & ( for n being Nat holds G . n = (F . n) | E ) holds for x being Element of X st x in E holds G # x is summable let S be SigmaField of X; ::_thesis: for E being Element of S for F, G being Functional_Sequence of X,ExtREAL st E c= dom (F . 0) & G is with_the_same_dom & ( for x being Element of X st x in E holds F # x is summable ) & ( for n being Nat holds G . n = (F . n) | E ) holds for x being Element of X st x in E holds G # x is summable let E be Element of S; ::_thesis: for F, G being Functional_Sequence of X,ExtREAL st E c= dom (F . 0) & G is with_the_same_dom & ( for x being Element of X st x in E holds F # x is summable ) & ( for n being Nat holds G . n = (F . n) | E ) holds for x being Element of X st x in E holds G # x is summable let F, G be Functional_Sequence of X,ExtREAL; ::_thesis: ( E c= dom (F . 0) & G is with_the_same_dom & ( for x being Element of X st x in E holds F # x is summable ) & ( for n being Nat holds G . n = (F . n) | E ) implies for x being Element of X st x in E holds G # x is summable ) assume that A1: E c= dom (F . 0) and A2: G is with_the_same_dom and A3: for x being Element of X st x in E holds F # x is summable and A4: for n being Nat holds G . n = (F . n) | E ; ::_thesis: for x being Element of X st x in E holds G # x is summable let x be Element of X; ::_thesis: ( x in E implies G # x is summable ) assume A5: x in E ; ::_thesis: G # x is summable dom ((F . 0) | E) = E by A1, RELAT_1:62; then A6: E = dom (G . 0) by A4; for n being Element of NAT holds (F # x) . n = (G # x) . n proof let n be Element of NAT ; ::_thesis: (F # x) . n = (G # x) . n dom (G . n) = E by A2, A6, MESFUNC8:def_2; then x in dom ((F . n) | E) by A4, A5; then ((F . n) | E) . x = (F . n) . x by FUNCT_1:47; then A7: (G . n) . x = (F . n) . x by A4; (F # x) . n = (F . n) . x by MESFUNC5:def_13; hence (F # x) . n = (G # x) . n by A7, MESFUNC5:def_13; ::_thesis: verum end; then A8: Partial_Sums (F # x) = Partial_Sums (G # x) by FUNCT_2:63; F # x is summable by A3, A5; then Partial_Sums (F # x) is convergent by Def2; hence G # x is summable by A8, Def2; ::_thesis: verum end; begin definition let X be non empty set ; let F be Functional_Sequence of X,ExtREAL; func Partial_Sums F -> Functional_Sequence of X,ExtREAL means :Def4: :: MESFUNC9:def 4 ( it . 0 = F . 0 & ( for n being Nat holds it . (n + 1) = (it . n) + (F . (n + 1)) ) ); existence ex b1 being Functional_Sequence of X,ExtREAL st ( b1 . 0 = F . 0 & ( for n being Nat holds b1 . (n + 1) = (b1 . n) + (F . (n + 1)) ) ) proof defpred S1[ Element of NAT , set , set ] means ( ( $2 is not PartFunc of X,ExtREAL & $3 = F . $1 ) or ( $2 is PartFunc of X,ExtREAL & ( for F2 being PartFunc of X,ExtREAL st F2 = $2 holds $3 = F2 + (F . ($1 + 1)) ) ) ); A1: for n being Element of NAT for x being set ex y being set st S1[n,x,y] proof let n be Element of NAT ; ::_thesis: for x being set ex y being set st S1[n,x,y] let x be set ; ::_thesis: ex y being set st S1[n,x,y] thus ex y being set st S1[n,x,y] ::_thesis: verum proof percases ( not x is PartFunc of X,ExtREAL or x is PartFunc of X,ExtREAL ) ; supposeA2: x is not PartFunc of X,ExtREAL ; ::_thesis: ex y being set st S1[n,x,y] take y = F . n; ::_thesis: S1[n,x,y] thus ( ( x is not PartFunc of X,ExtREAL & y = F . n ) or ( x is PartFunc of X,ExtREAL & ( for F2 being PartFunc of X,ExtREAL st F2 = x holds y = F2 + (F . (n + 1)) ) ) ) by A2; ::_thesis: verum end; suppose x is PartFunc of X,ExtREAL ; ::_thesis: ex y being set st S1[n,x,y] then reconsider G2 = x as PartFunc of X,ExtREAL ; take y = G2 + (F . (n + 1)); ::_thesis: S1[n,x,y] thus ( ( x is not PartFunc of X,ExtREAL & y = F . n ) or ( x is PartFunc of X,ExtREAL & ( for F2 being PartFunc of X,ExtREAL st F2 = x holds y = F2 + (F . (n + 1)) ) ) ) ; ::_thesis: verum end; end; end; end; consider IT being Function such that A3: ( dom IT = NAT & IT . 0 = F . 0 & ( for n being Element of NAT holds S1[n,IT . n,IT . (n + 1)] ) ) from RECDEF_1:sch_1(A1); now__::_thesis:_for_f_being_set_st_f_in_rng_IT_holds_ f_in_PFuncs_(X,ExtREAL) defpred S2[ Element of NAT ] means IT . $1 is PartFunc of X,ExtREAL; let f be set ; ::_thesis: ( f in rng IT implies f in PFuncs (X,ExtREAL) ) assume f in rng IT ; ::_thesis: f in PFuncs (X,ExtREAL) then consider m being set such that A4: m in dom IT and A5: f = IT . m by FUNCT_1:def_3; reconsider m = m as Element of NAT by A3, A4; A6: for n being Element of NAT st S2[n] holds S2[n + 1] proof let n be Element of NAT ; ::_thesis: ( S2[n] implies S2[n + 1] ) assume S2[n] ; ::_thesis: S2[n + 1] then reconsider F2 = IT . n as PartFunc of X,ExtREAL ; IT . (n + 1) = F2 + (F . (n + 1)) by A3; hence S2[n + 1] ; ::_thesis: verum end; A7: S2[ 0 ] by A3; for n being Element of NAT holds S2[n] from NAT_1:sch_1(A7, A6); then IT . m is PartFunc of X,ExtREAL ; hence f in PFuncs (X,ExtREAL) by A5, PARTFUN1:45; ::_thesis: verum end; then rng IT c= PFuncs (X,ExtREAL) by TARSKI:def_3; then reconsider IT = IT as Functional_Sequence of X,ExtREAL by A3, FUNCT_2:def_1, RELSET_1:4; take IT ; ::_thesis: ( IT . 0 = F . 0 & ( for n being Nat holds IT . (n + 1) = (IT . n) + (F . (n + 1)) ) ) for n being Nat holds IT . (n + 1) = (IT . n) + (F . (n + 1)) proof let n be Nat; ::_thesis: IT . (n + 1) = (IT . n) + (F . (n + 1)) reconsider m = n as Element of NAT by ORDINAL1:def_12; IT . (m + 1) = (IT . m) + (F . (m + 1)) by A3; hence IT . (n + 1) = (IT . n) + (F . (n + 1)) ; ::_thesis: verum end; hence ( IT . 0 = F . 0 & ( for n being Nat holds IT . (n + 1) = (IT . n) + (F . (n + 1)) ) ) by A3; ::_thesis: verum end; uniqueness for b1, b2 being Functional_Sequence of X,ExtREAL st b1 . 0 = F . 0 & ( for n being Nat holds b1 . (n + 1) = (b1 . n) + (F . (n + 1)) ) & b2 . 0 = F . 0 & ( for n being Nat holds b2 . (n + 1) = (b2 . n) + (F . (n + 1)) ) holds b1 = b2 proof let PS1, PS2 be Functional_Sequence of X,ExtREAL; ::_thesis: ( PS1 . 0 = F . 0 & ( for n being Nat holds PS1 . (n + 1) = (PS1 . n) + (F . (n + 1)) ) & PS2 . 0 = F . 0 & ( for n being Nat holds PS2 . (n + 1) = (PS2 . n) + (F . (n + 1)) ) implies PS1 = PS2 ) assume that A8: PS1 . 0 = F . 0 and A9: for n being Nat holds PS1 . (n + 1) = (PS1 . n) + (F . (n + 1)) and A10: PS2 . 0 = F . 0 and A11: for n being Nat holds PS2 . (n + 1) = (PS2 . n) + (F . (n + 1)) ; ::_thesis: PS1 = PS2 defpred S1[ Nat] means PS1 . $1 = PS2 . $1; A12: for n being Nat st S1[n] holds S1[n + 1] proof let n be Nat; ::_thesis: ( S1[n] implies S1[n + 1] ) assume A13: S1[n] ; ::_thesis: S1[n + 1] PS1 . (n + 1) = (PS1 . n) + (F . (n + 1)) by A9; hence S1[n + 1] by A11, A13; ::_thesis: verum end; A14: S1[ 0 ] by A8, A10; for n being Nat holds S1[n] from NAT_1:sch_2(A14, A12); then for m being Element of NAT holds PS1 . m = PS2 . m ; hence PS1 = PS2 by FUNCT_2:63; ::_thesis: verum end; end; :: deftheorem Def4 defines Partial_Sums MESFUNC9:def_4_:_ for X being non empty set for F, b3 being Functional_Sequence of X,ExtREAL holds ( b3 = Partial_Sums F iff ( b3 . 0 = F . 0 & ( for n being Nat holds b3 . (n + 1) = (b3 . n) + (F . (n + 1)) ) ) ); definition let X be set ; let F be Functional_Sequence of X,ExtREAL; attrF is additive means :Def5: :: MESFUNC9:def 5 for n, m being Nat st n <> m holds for x being set holds ( not x in (dom (F . n)) /\ (dom (F . m)) or (F . n) . x <> +infty or (F . m) . x <> -infty ); end; :: deftheorem Def5 defines additive MESFUNC9:def_5_:_ for X being set for F being Functional_Sequence of X,ExtREAL holds ( F is additive iff for n, m being Nat st n <> m holds for x being set holds ( not x in (dom (F . n)) /\ (dom (F . m)) or (F . n) . x <> +infty or (F . m) . x <> -infty ) ); Lm1: for X being non empty set for F being Functional_Sequence of X,ExtREAL for n, m being Nat for z being set st z in dom ((Partial_Sums F) . n) & m <= n holds z in dom ((Partial_Sums F) . m) proof let X be non empty set ; ::_thesis: for F being Functional_Sequence of X,ExtREAL for n, m being Nat for z being set st z in dom ((Partial_Sums F) . n) & m <= n holds z in dom ((Partial_Sums F) . m) let F be Functional_Sequence of X,ExtREAL; ::_thesis: for n, m being Nat for z being set st z in dom ((Partial_Sums F) . n) & m <= n holds z in dom ((Partial_Sums F) . m) let n, m be Nat; ::_thesis: for z being set st z in dom ((Partial_Sums F) . n) & m <= n holds z in dom ((Partial_Sums F) . m) let z be set ; ::_thesis: ( z in dom ((Partial_Sums F) . n) & m <= n implies z in dom ((Partial_Sums F) . m) ) set PF = Partial_Sums F; assume that A1: z in dom ((Partial_Sums F) . n) and A2: m <= n ; ::_thesis: z in dom ((Partial_Sums F) . m) defpred S1[ Nat] means ( m <= $1 & $1 <= n implies not z in dom ((Partial_Sums F) . $1) ); assume A3: not z in dom ((Partial_Sums F) . m) ; ::_thesis: contradiction A4: for k being Nat st S1[k] holds S1[k + 1] proof let k be Nat; ::_thesis: ( S1[k] implies S1[k + 1] ) assume A5: S1[k] ; ::_thesis: S1[k + 1] assume that A6: m <= k + 1 and A7: k + 1 <= n ; ::_thesis: not z in dom ((Partial_Sums F) . (k + 1)) percases ( m <= k or m = k + 1 ) by A6, NAT_1:8; supposeA8: m <= k ; ::_thesis: not z in dom ((Partial_Sums F) . (k + 1)) (Partial_Sums F) . (k + 1) = ((Partial_Sums F) . k) + (F . (k + 1)) by Def4; then A9: dom ((Partial_Sums F) . (k + 1)) = ((dom ((Partial_Sums F) . k)) /\ (dom (F . (k + 1)))) \ (((((Partial_Sums F) . k) " {-infty}) /\ ((F . (k + 1)) " {+infty})) \/ ((((Partial_Sums F) . k) " {+infty}) /\ ((F . (k + 1)) " {-infty}))) by MESFUNC1:def_3; not z in (dom ((Partial_Sums F) . k)) /\ (dom (F . (k + 1))) by A5, A7, A8, NAT_1:13, XBOOLE_0:def_4; hence not z in dom ((Partial_Sums F) . (k + 1)) by A9, XBOOLE_0:def_5; ::_thesis: verum end; suppose m = k + 1 ; ::_thesis: not z in dom ((Partial_Sums F) . (k + 1)) hence not z in dom ((Partial_Sums F) . (k + 1)) by A3; ::_thesis: verum end; end; end; A10: S1[ 0 ] by A3; for k being Nat holds S1[k] from NAT_1:sch_2(A10, A4); hence contradiction by A1, A2; ::_thesis: verum end; theorem Th22: :: MESFUNC9:22 for X being non empty set for F being Functional_Sequence of X,ExtREAL for n, m being Nat for z being set st z in dom ((Partial_Sums F) . n) & m <= n holds ( z in dom ((Partial_Sums F) . m) & z in dom (F . m) ) proof let X be non empty set ; ::_thesis: for F being Functional_Sequence of X,ExtREAL for n, m being Nat for z being set st z in dom ((Partial_Sums F) . n) & m <= n holds ( z in dom ((Partial_Sums F) . m) & z in dom (F . m) ) let F be Functional_Sequence of X,ExtREAL; ::_thesis: for n, m being Nat for z being set st z in dom ((Partial_Sums F) . n) & m <= n holds ( z in dom ((Partial_Sums F) . m) & z in dom (F . m) ) let n, m be Nat; ::_thesis: for z being set st z in dom ((Partial_Sums F) . n) & m <= n holds ( z in dom ((Partial_Sums F) . m) & z in dom (F . m) ) let z be set ; ::_thesis: ( z in dom ((Partial_Sums F) . n) & m <= n implies ( z in dom ((Partial_Sums F) . m) & z in dom (F . m) ) ) set PF = Partial_Sums F; assume that A1: z in dom ((Partial_Sums F) . n) and A2: m <= n ; ::_thesis: ( z in dom ((Partial_Sums F) . m) & z in dom (F . m) ) thus A3: z in dom ((Partial_Sums F) . m) by A1, A2, Lm1; ::_thesis: z in dom (F . m) percases ( m = 0 or m <> 0 ) ; suppose m = 0 ; ::_thesis: z in dom (F . m) then (Partial_Sums F) . m = F . m by Def4; hence z in dom (F . m) by A1, A2, Lm1; ::_thesis: verum end; suppose m <> 0 ; ::_thesis: z in dom (F . m) then consider k being Nat such that A4: m = k + 1 by NAT_1:6; (Partial_Sums F) . m = ((Partial_Sums F) . k) + (F . m) by A4, Def4; then dom ((Partial_Sums F) . m) = ((dom ((Partial_Sums F) . k)) /\ (dom (F . m))) \ (((((Partial_Sums F) . k) " {-infty}) /\ ((F . m) " {+infty})) \/ ((((Partial_Sums F) . k) " {+infty}) /\ ((F . m) " {-infty}))) by MESFUNC1:def_3; then z in (dom ((Partial_Sums F) . k)) /\ (dom (F . m)) by A3, XBOOLE_0:def_5; hence z in dom (F . m) by XBOOLE_0:def_4; ::_thesis: verum end; end; end; theorem Th23: :: MESFUNC9:23 for X being non empty set for F being Functional_Sequence of X,ExtREAL for n being Nat for z being set st z in dom ((Partial_Sums F) . n) & ((Partial_Sums F) . n) . z = +infty holds ex m being Nat st ( m <= n & (F . m) . z = +infty ) proof let X be non empty set ; ::_thesis: for F being Functional_Sequence of X,ExtREAL for n being Nat for z being set st z in dom ((Partial_Sums F) . n) & ((Partial_Sums F) . n) . z = +infty holds ex m being Nat st ( m <= n & (F . m) . z = +infty ) let F be Functional_Sequence of X,ExtREAL; ::_thesis: for n being Nat for z being set st z in dom ((Partial_Sums F) . n) & ((Partial_Sums F) . n) . z = +infty holds ex m being Nat st ( m <= n & (F . m) . z = +infty ) let n be Nat; ::_thesis: for z being set st z in dom ((Partial_Sums F) . n) & ((Partial_Sums F) . n) . z = +infty holds ex m being Nat st ( m <= n & (F . m) . z = +infty ) let z be set ; ::_thesis: ( z in dom ((Partial_Sums F) . n) & ((Partial_Sums F) . n) . z = +infty implies ex m being Nat st ( m <= n & (F . m) . z = +infty ) ) set PF = Partial_Sums F; assume that A1: z in dom ((Partial_Sums F) . n) and A2: ((Partial_Sums F) . n) . z = +infty ; ::_thesis: ex m being Nat st ( m <= n & (F . m) . z = +infty ) now__::_thesis:_ex_m_being_Element_of_NAT_st_ (_m_<=_n_&_not_(F_._m)_._z_<>_+infty_) defpred S1[ Nat] means ( $1 <= n implies ((Partial_Sums F) . $1) . z <> +infty ); assume A3: for m being Element of NAT st m <= n holds (F . m) . z <> +infty ; ::_thesis: contradiction A4: for k being Nat st S1[k] holds S1[k + 1] proof let k be Nat; ::_thesis: ( S1[k] implies S1[k + 1] ) assume A5: S1[k] ; ::_thesis: S1[k + 1] assume A6: k + 1 <= n ; ::_thesis: ((Partial_Sums F) . (k + 1)) . z <> +infty then k <= n by NAT_1:13; then A7: z in dom ((Partial_Sums F) . k) by A1, Th22; not ((Partial_Sums F) . k) . z in {+infty} by A5, A6, NAT_1:13, TARSKI:def_1; then not z in ((Partial_Sums F) . k) " {+infty} by FUNCT_1:def_7; then A8: not z in (((Partial_Sums F) . k) " {+infty}) /\ ((F . (k + 1)) " {-infty}) by XBOOLE_0:def_4; z in dom (F . (k + 1)) by A1, A6, Th22; then A9: z in (dom ((Partial_Sums F) . k)) /\ (dom (F . (k + 1))) by A7, XBOOLE_0:def_4; A10: (Partial_Sums F) . (k + 1) = ((Partial_Sums F) . k) + (F . (k + 1)) by Def4; A11: (F . (k + 1)) . z <> +infty by A3, A6; then not (F . (k + 1)) . z in {+infty} by TARSKI:def_1; then not z in (F . (k + 1)) " {+infty} by FUNCT_1:def_7; then not z in (((Partial_Sums F) . k) " {-infty}) /\ ((F . (k + 1)) " {+infty}) by XBOOLE_0:def_4; then not z in ((((Partial_Sums F) . k) " {+infty}) /\ ((F . (k + 1)) " {-infty})) \/ ((((Partial_Sums F) . k) " {-infty}) /\ ((F . (k + 1)) " {+infty})) by A8, XBOOLE_0:def_3; then z in ((dom ((Partial_Sums F) . k)) /\ (dom (F . (k + 1)))) \ (((((Partial_Sums F) . k) " {+infty}) /\ ((F . (k + 1)) " {-infty})) \/ ((((Partial_Sums F) . k) " {-infty}) /\ ((F . (k + 1)) " {+infty}))) by A9, XBOOLE_0:def_5; then z in dom ((Partial_Sums F) . (k + 1)) by A10, MESFUNC1:def_3; then ((Partial_Sums F) . (k + 1)) . z = (((Partial_Sums F) . k) . z) + ((F . (k + 1)) . z) by A10, MESFUNC1:def_3; hence ((Partial_Sums F) . (k + 1)) . z <> +infty by A5, A6, A11, NAT_1:13, XXREAL_3:16; ::_thesis: verum end; (Partial_Sums F) . 0 = F . 0 by Def4; then A12: S1[ 0 ] by A3; for k being Nat holds S1[k] from NAT_1:sch_2(A12, A4); hence contradiction by A2; ::_thesis: verum end; hence ex m being Nat st ( m <= n & (F . m) . z = +infty ) ; ::_thesis: verum end; theorem :: MESFUNC9:24 for X being non empty set for F being Functional_Sequence of X,ExtREAL for n, m being Nat for z being set st F is additive & z in dom ((Partial_Sums F) . n) & ((Partial_Sums F) . n) . z = +infty & m <= n holds (F . m) . z <> -infty proof let X be non empty set ; ::_thesis: for F being Functional_Sequence of X,ExtREAL for n, m being Nat for z being set st F is additive & z in dom ((Partial_Sums F) . n) & ((Partial_Sums F) . n) . z = +infty & m <= n holds (F . m) . z <> -infty let F be Functional_Sequence of X,ExtREAL; ::_thesis: for n, m being Nat for z being set st F is additive & z in dom ((Partial_Sums F) . n) & ((Partial_Sums F) . n) . z = +infty & m <= n holds (F . m) . z <> -infty let n, m be Nat; ::_thesis: for z being set st F is additive & z in dom ((Partial_Sums F) . n) & ((Partial_Sums F) . n) . z = +infty & m <= n holds (F . m) . z <> -infty let z be set ; ::_thesis: ( F is additive & z in dom ((Partial_Sums F) . n) & ((Partial_Sums F) . n) . z = +infty & m <= n implies (F . m) . z <> -infty ) assume that A1: F is additive and A2: z in dom ((Partial_Sums F) . n) and A3: ((Partial_Sums F) . n) . z = +infty and A4: m <= n ; ::_thesis: (F . m) . z <> -infty A5: z in dom (F . m) by A2, A4, Th22; consider k being Nat such that A6: k <= n and A7: (F . k) . z = +infty by A2, A3, Th23; z in dom (F . k) by A2, A6, Th22; then z in (dom (F . m)) /\ (dom (F . k)) by A5, XBOOLE_0:def_4; hence (F . m) . z <> -infty by A1, A7, Def5; ::_thesis: verum end; theorem Th25: :: MESFUNC9:25 for X being non empty set for F being Functional_Sequence of X,ExtREAL for n being Nat for z being set st z in dom ((Partial_Sums F) . n) & ((Partial_Sums F) . n) . z = -infty holds ex m being Nat st ( m <= n & (F . m) . z = -infty ) proof let X be non empty set ; ::_thesis: for F being Functional_Sequence of X,ExtREAL for n being Nat for z being set st z in dom ((Partial_Sums F) . n) & ((Partial_Sums F) . n) . z = -infty holds ex m being Nat st ( m <= n & (F . m) . z = -infty ) let F be Functional_Sequence of X,ExtREAL; ::_thesis: for n being Nat for z being set st z in dom ((Partial_Sums F) . n) & ((Partial_Sums F) . n) . z = -infty holds ex m being Nat st ( m <= n & (F . m) . z = -infty ) let n be Nat; ::_thesis: for z being set st z in dom ((Partial_Sums F) . n) & ((Partial_Sums F) . n) . z = -infty holds ex m being Nat st ( m <= n & (F . m) . z = -infty ) let z be set ; ::_thesis: ( z in dom ((Partial_Sums F) . n) & ((Partial_Sums F) . n) . z = -infty implies ex m being Nat st ( m <= n & (F . m) . z = -infty ) ) set PF = Partial_Sums F; assume that A1: z in dom ((Partial_Sums F) . n) and A2: ((Partial_Sums F) . n) . z = -infty ; ::_thesis: ex m being Nat st ( m <= n & (F . m) . z = -infty ) now__::_thesis:_ex_m_being_Nat_st_ (_m_<=_n_&_not_(F_._m)_._z_<>_-infty_) defpred S1[ Nat] means ( $1 <= n implies ((Partial_Sums F) . $1) . z <> -infty ); assume A3: for m being Nat st m <= n holds (F . m) . z <> -infty ; ::_thesis: contradiction A4: for k being Nat st S1[k] holds S1[k + 1] proof let k be Nat; ::_thesis: ( S1[k] implies S1[k + 1] ) assume A5: S1[k] ; ::_thesis: S1[k + 1] assume A6: k + 1 <= n ; ::_thesis: ((Partial_Sums F) . (k + 1)) . z <> -infty then k <= n by NAT_1:13; then A7: z in dom ((Partial_Sums F) . k) by A1, Th22; not ((Partial_Sums F) . k) . z in {-infty} by A5, A6, NAT_1:13, TARSKI:def_1; then not z in ((Partial_Sums F) . k) " {-infty} by FUNCT_1:def_7; then A8: not z in (((Partial_Sums F) . k) " {-infty}) /\ ((F . (k + 1)) " {+infty}) by XBOOLE_0:def_4; z in dom (F . (k + 1)) by A1, A6, Th22; then A9: z in (dom ((Partial_Sums F) . k)) /\ (dom (F . (k + 1))) by A7, XBOOLE_0:def_4; A10: (Partial_Sums F) . (k + 1) = ((Partial_Sums F) . k) + (F . (k + 1)) by Def4; A11: (F . (k + 1)) . z <> -infty by A3, A6; then not (F . (k + 1)) . z in {-infty} by TARSKI:def_1; then not z in (F . (k + 1)) " {-infty} by FUNCT_1:def_7; then not z in (((Partial_Sums F) . k) " {+infty}) /\ ((F . (k + 1)) " {-infty}) by XBOOLE_0:def_4; then not z in ((((Partial_Sums F) . k) " {-infty}) /\ ((F . (k + 1)) " {+infty})) \/ ((((Partial_Sums F) . k) " {+infty}) /\ ((F . (k + 1)) " {-infty})) by A8, XBOOLE_0:def_3; then z in ((dom ((Partial_Sums F) . k)) /\ (dom (F . (k + 1)))) \ (((((Partial_Sums F) . k) " {-infty}) /\ ((F . (k + 1)) " {+infty})) \/ ((((Partial_Sums F) . k) " {+infty}) /\ ((F . (k + 1)) " {-infty}))) by A9, XBOOLE_0:def_5; then z in dom ((Partial_Sums F) . (k + 1)) by A10, MESFUNC1:def_3; then ((Partial_Sums F) . (k + 1)) . z = (((Partial_Sums F) . k) . z) + ((F . (k + 1)) . z) by A10, MESFUNC1:def_3; hence ((Partial_Sums F) . (k + 1)) . z <> -infty by A5, A6, A11, NAT_1:13, XXREAL_3:17; ::_thesis: verum end; (Partial_Sums F) . 0 = F . 0 by Def4; then A12: S1[ 0 ] by A3; for k being Nat holds S1[k] from NAT_1:sch_2(A12, A4); hence contradiction by A2; ::_thesis: verum end; hence ex m being Nat st ( m <= n & (F . m) . z = -infty ) ; ::_thesis: verum end; theorem :: MESFUNC9:26 for X being non empty set for F being Functional_Sequence of X,ExtREAL for n, m being Nat for z being set st F is additive & z in dom ((Partial_Sums F) . n) & ((Partial_Sums F) . n) . z = -infty & m <= n holds (F . m) . z <> +infty proof let X be non empty set ; ::_thesis: for F being Functional_Sequence of X,ExtREAL for n, m being Nat for z being set st F is additive & z in dom ((Partial_Sums F) . n) & ((Partial_Sums F) . n) . z = -infty & m <= n holds (F . m) . z <> +infty let F be Functional_Sequence of X,ExtREAL; ::_thesis: for n, m being Nat for z being set st F is additive & z in dom ((Partial_Sums F) . n) & ((Partial_Sums F) . n) . z = -infty & m <= n holds (F . m) . z <> +infty let n, m be Nat; ::_thesis: for z being set st F is additive & z in dom ((Partial_Sums F) . n) & ((Partial_Sums F) . n) . z = -infty & m <= n holds (F . m) . z <> +infty let z be set ; ::_thesis: ( F is additive & z in dom ((Partial_Sums F) . n) & ((Partial_Sums F) . n) . z = -infty & m <= n implies (F . m) . z <> +infty ) assume A1: F is additive ; ::_thesis: ( not z in dom ((Partial_Sums F) . n) or not ((Partial_Sums F) . n) . z = -infty or not m <= n or (F . m) . z <> +infty ) assume that A2: z in dom ((Partial_Sums F) . n) and A3: ((Partial_Sums F) . n) . z = -infty ; ::_thesis: ( not m <= n or (F . m) . z <> +infty ) assume m <= n ; ::_thesis: (F . m) . z <> +infty then A4: z in dom (F . m) by A2, Th22; consider k being Nat such that A5: k <= n and A6: (F . k) . z = -infty by A2, A3, Th25; z in dom (F . k) by A2, A5, Th22; then z in (dom (F . m)) /\ (dom (F . k)) by A4, XBOOLE_0:def_4; hence (F . m) . z <> +infty by A1, A6, Def5; ::_thesis: verum end; theorem Th27: :: MESFUNC9:27 for X being non empty set for F being Functional_Sequence of X,ExtREAL for n being Nat st F is additive holds ( (((Partial_Sums F) . n) " {-infty}) /\ ((F . (n + 1)) " {+infty}) = {} & (((Partial_Sums F) . n) " {+infty}) /\ ((F . (n + 1)) " {-infty}) = {} ) proof let X be non empty set ; ::_thesis: for F being Functional_Sequence of X,ExtREAL for n being Nat st F is additive holds ( (((Partial_Sums F) . n) " {-infty}) /\ ((F . (n + 1)) " {+infty}) = {} & (((Partial_Sums F) . n) " {+infty}) /\ ((F . (n + 1)) " {-infty}) = {} ) let F be Functional_Sequence of X,ExtREAL; ::_thesis: for n being Nat st F is additive holds ( (((Partial_Sums F) . n) " {-infty}) /\ ((F . (n + 1)) " {+infty}) = {} & (((Partial_Sums F) . n) " {+infty}) /\ ((F . (n + 1)) " {-infty}) = {} ) let n be Nat; ::_thesis: ( F is additive implies ( (((Partial_Sums F) . n) " {-infty}) /\ ((F . (n + 1)) " {+infty}) = {} & (((Partial_Sums F) . n) " {+infty}) /\ ((F . (n + 1)) " {-infty}) = {} ) ) set PF = Partial_Sums F; assume A1: F is additive ; ::_thesis: ( (((Partial_Sums F) . n) " {-infty}) /\ ((F . (n + 1)) " {+infty}) = {} & (((Partial_Sums F) . n) " {+infty}) /\ ((F . (n + 1)) " {-infty}) = {} ) now__::_thesis:_for_x_being_set_holds_ (_not_x_in_((Partial_Sums_F)_._n)_"_{-infty}_or_not_x_in_(F_._(n_+_1))_"_{+infty}_) assume ex x being set st ( x in ((Partial_Sums F) . n) " {-infty} & x in (F . (n + 1)) " {+infty} ) ; ::_thesis: contradiction then consider z being set such that A2: z in ((Partial_Sums F) . n) " {-infty} and A3: z in (F . (n + 1)) " {+infty} ; A4: z in dom ((Partial_Sums F) . n) by A2, FUNCT_1:def_7; ((Partial_Sums F) . n) . z in {-infty} by A2, FUNCT_1:def_7; then ((Partial_Sums F) . n) . z = -infty by TARSKI:def_1; then consider k being Nat such that A5: k <= n and A6: (F . k) . z = -infty by A4, Th25; A7: z in dom (F . (n + 1)) by A3, FUNCT_1:def_7; (F . (n + 1)) . z in {+infty} by A3, FUNCT_1:def_7; then A8: (F . (n + 1)) . z = +infty by TARSKI:def_1; z in dom (F . k) by A4, A5, Th22; then z in (dom (F . k)) /\ (dom (F . (n + 1))) by A7, XBOOLE_0:def_4; hence contradiction by A1, A8, A6, Def5; ::_thesis: verum end; then ((Partial_Sums F) . n) " {-infty} misses (F . (n + 1)) " {+infty} by XBOOLE_0:3; hence (((Partial_Sums F) . n) " {-infty}) /\ ((F . (n + 1)) " {+infty}) = {} by XBOOLE_0:def_7; ::_thesis: (((Partial_Sums F) . n) " {+infty}) /\ ((F . (n + 1)) " {-infty}) = {} now__::_thesis:_for_x_being_set_holds_ (_not_x_in_((Partial_Sums_F)_._n)_"_{+infty}_or_not_x_in_(F_._(n_+_1))_"_{-infty}_) assume ex x being set st ( x in ((Partial_Sums F) . n) " {+infty} & x in (F . (n + 1)) " {-infty} ) ; ::_thesis: contradiction then consider z being set such that A9: z in ((Partial_Sums F) . n) " {+infty} and A10: z in (F . (n + 1)) " {-infty} ; A11: z in dom ((Partial_Sums F) . n) by A9, FUNCT_1:def_7; ((Partial_Sums F) . n) . z in {+infty} by A9, FUNCT_1:def_7; then ((Partial_Sums F) . n) . z = +infty by TARSKI:def_1; then consider k being Nat such that A12: k <= n and A13: (F . k) . z = +infty by A11, Th23; A14: z in dom (F . (n + 1)) by A10, FUNCT_1:def_7; (F . (n + 1)) . z in {-infty} by A10, FUNCT_1:def_7; then A15: (F . (n + 1)) . z = -infty by TARSKI:def_1; z in dom (F . k) by A11, A12, Th22; then z in (dom (F . k)) /\ (dom (F . (n + 1))) by A14, XBOOLE_0:def_4; hence contradiction by A1, A15, A13, Def5; ::_thesis: verum end; then ((Partial_Sums F) . n) " {+infty} misses (F . (n + 1)) " {-infty} by XBOOLE_0:3; hence (((Partial_Sums F) . n) " {+infty}) /\ ((F . (n + 1)) " {-infty}) = {} by XBOOLE_0:def_7; ::_thesis: verum end; theorem Th28: :: MESFUNC9:28 for X being non empty set for F being Functional_Sequence of X,ExtREAL for n being Nat st F is additive holds dom ((Partial_Sums F) . n) = meet { (dom (F . k)) where k is Element of NAT : k <= n } proof let X be non empty set ; ::_thesis: for F being Functional_Sequence of X,ExtREAL for n being Nat st F is additive holds dom ((Partial_Sums F) . n) = meet { (dom (F . k)) where k is Element of NAT : k <= n } let F be Functional_Sequence of X,ExtREAL; ::_thesis: for n being Nat st F is additive holds dom ((Partial_Sums F) . n) = meet { (dom (F . k)) where k is Element of NAT : k <= n } let n be Nat; ::_thesis: ( F is additive implies dom ((Partial_Sums F) . n) = meet { (dom (F . k)) where k is Element of NAT : k <= n } ) deffunc H1( Nat) -> set = { (dom (F . k)) where k is Element of NAT : k <= $1 } ; set PF = Partial_Sums F; defpred S1[ Nat] means dom ((Partial_Sums F) . $1) = meet { (dom (F . k)) where k is Element of NAT : k <= $1 } ; A1: dom ((Partial_Sums F) . 0) = dom (F . 0) by Def4; now__::_thesis:_for_V_being_set_st_V_in_H1(_0_)_holds_ V_in_{(dom_(F_._0))} let V be set ; ::_thesis: ( V in H1( 0 ) implies V in {(dom (F . 0))} ) assume V in H1( 0 ) ; ::_thesis: V in {(dom (F . 0))} then consider k being Element of NAT such that A2: V = dom (F . k) and A3: k <= 0 ; k = 0 by A3; hence V in {(dom (F . 0))} by A2, TARSKI:def_1; ::_thesis: verum end; then A4: H1( 0 ) c= {(dom (F . 0))} by TARSKI:def_3; assume A5: F is additive ; ::_thesis: dom ((Partial_Sums F) . n) = meet { (dom (F . k)) where k is Element of NAT : k <= n } A6: for m being Nat st S1[m] holds S1[m + 1] proof let m be Nat; ::_thesis: ( S1[m] implies S1[m + 1] ) A7: (Partial_Sums F) . (m + 1) = ((Partial_Sums F) . m) + (F . (m + 1)) by Def4; A8: (((Partial_Sums F) . m) " {+infty}) /\ ((F . (m + 1)) " {-infty}) = {} by A5, Th27; A9: dom (F . 0) in H1(m + 1) ; now__::_thesis:_for_V_being_set_st_V_in_(meet_H1(m))_/\_(dom_(F_._(m_+_1)))_holds_ V_in_meet_H1(m_+_1) let V be set ; ::_thesis: ( V in (meet H1(m)) /\ (dom (F . (m + 1))) implies V in meet H1(m + 1) ) assume A10: V in (meet H1(m)) /\ (dom (F . (m + 1))) ; ::_thesis: V in meet H1(m + 1) then A11: V in dom (F . (m + 1)) by XBOOLE_0:def_4; A12: V in meet H1(m) by A10, XBOOLE_0:def_4; for W being set st W in H1(m + 1) holds V in W proof let W be set ; ::_thesis: ( W in H1(m + 1) implies V in W ) assume W in H1(m + 1) ; ::_thesis: V in W then consider i being Element of NAT such that A13: W = dom (F . i) and A14: i <= m + 1 ; now__::_thesis:_(_i_<=_m_implies_V_in_W_) assume i <= m ; ::_thesis: V in W then W in H1(m) by A13; hence V in W by A12, SETFAM_1:def_1; ::_thesis: verum end; hence V in W by A11, A13, A14, NAT_1:8; ::_thesis: verum end; hence V in meet H1(m + 1) by A9, SETFAM_1:def_1; ::_thesis: verum end; then A15: (meet H1(m)) /\ (dom (F . (m + 1))) c= meet H1(m + 1) by TARSKI:def_3; A16: dom (F . 0) in H1(m) ; now__::_thesis:_for_V_being_set_st_V_in_meet_H1(m_+_1)_holds_ V_in_(meet_H1(m))_/\_(dom_(F_._(m_+_1))) now__::_thesis:_for_V_being_set_st_V_in_H1(m)_holds_ V_in_H1(m_+_1) let V be set ; ::_thesis: ( V in H1(m) implies V in H1(m + 1) ) assume V in H1(m) ; ::_thesis: V in H1(m + 1) then consider i being Element of NAT such that A17: V = dom (F . i) and A18: i <= m ; i <= m + 1 by A18, NAT_1:12; hence V in H1(m + 1) by A17; ::_thesis: verum end; then H1(m) c= H1(m + 1) by TARSKI:def_3; then A19: meet H1(m + 1) c= meet H1(m) by A16, SETFAM_1:6; let V be set ; ::_thesis: ( V in meet H1(m + 1) implies V in (meet H1(m)) /\ (dom (F . (m + 1))) ) assume A20: V in meet H1(m + 1) ; ::_thesis: V in (meet H1(m)) /\ (dom (F . (m + 1))) dom (F . (m + 1)) in H1(m + 1) ; then V in dom (F . (m + 1)) by A20, SETFAM_1:def_1; hence V in (meet H1(m)) /\ (dom (F . (m + 1))) by A20, A19, XBOOLE_0:def_4; ::_thesis: verum end; then A21: meet H1(m + 1) c= (meet H1(m)) /\ (dom (F . (m + 1))) by TARSKI:def_3; (((Partial_Sums F) . m) " {-infty}) /\ ((F . (m + 1)) " {+infty}) = {} by A5, Th27; then A22: dom ((Partial_Sums F) . (m + 1)) = ((dom ((Partial_Sums F) . m)) /\ (dom (F . (m + 1)))) \ ({} \/ {}) by A8, A7, MESFUNC1:def_3; assume S1[m] ; ::_thesis: S1[m + 1] hence S1[m + 1] by A22, A21, A15, XBOOLE_0:def_10; ::_thesis: verum end; now__::_thesis:_for_V_being_set_st_V_in_{(dom_(F_._0))}_holds_ V_in_H1(_0_) let V be set ; ::_thesis: ( V in {(dom (F . 0))} implies V in H1( 0 ) ) assume V in {(dom (F . 0))} ; ::_thesis: V in H1( 0 ) then V = dom (F . 0) by TARSKI:def_1; hence V in H1( 0 ) ; ::_thesis: verum end; then {(dom (F . 0))} c= H1( 0 ) by TARSKI:def_3; then H1( 0 ) = {(dom (F . 0))} by A4, XBOOLE_0:def_10; then A23: S1[ 0 ] by A1, SETFAM_1:10; for k being Nat holds S1[k] from NAT_1:sch_2(A23, A6); hence dom ((Partial_Sums F) . n) = meet { (dom (F . k)) where k is Element of NAT : k <= n } ; ::_thesis: verum end; theorem Th29: :: MESFUNC9:29 for X being non empty set for F being Functional_Sequence of X,ExtREAL for n being Nat st F is additive & F is with_the_same_dom holds dom ((Partial_Sums F) . n) = dom (F . 0) proof let X be non empty set ; ::_thesis: for F being Functional_Sequence of X,ExtREAL for n being Nat st F is additive & F is with_the_same_dom holds dom ((Partial_Sums F) . n) = dom (F . 0) let F be Functional_Sequence of X,ExtREAL; ::_thesis: for n being Nat st F is additive & F is with_the_same_dom holds dom ((Partial_Sums F) . n) = dom (F . 0) let n be Nat; ::_thesis: ( F is additive & F is with_the_same_dom implies dom ((Partial_Sums F) . n) = dom (F . 0) ) assume that A1: F is additive and A2: F is with_the_same_dom ; ::_thesis: dom ((Partial_Sums F) . n) = dom (F . 0) now__::_thesis:_for_D_being_set_st_D_in_meet__{__(dom_(F_._k))_where_k_is_Element_of_NAT_:_k_<=_n__}__holds_ D_in_meet_{(dom_(F_._0))} let D be set ; ::_thesis: ( D in meet { (dom (F . k)) where k is Element of NAT : k <= n } implies D in meet {(dom (F . 0))} ) A3: dom (F . 0) in { (dom (F . k)) where k is Element of NAT : k <= n } ; assume D in meet { (dom (F . k)) where k is Element of NAT : k <= n } ; ::_thesis: D in meet {(dom (F . 0))} then D in dom (F . 0) by A3, SETFAM_1:def_1; hence D in meet {(dom (F . 0))} by SETFAM_1:10; ::_thesis: verum end; then A4: meet { (dom (F . k)) where k is Element of NAT : k <= n } c= meet {(dom (F . 0))} by TARSKI:def_3; now__::_thesis:_for_D_being_set_st_D_in_meet_{(dom_(F_._0))}_holds_ D_in_meet__{__(dom_(F_._k))_where_k_is_Element_of_NAT_:_k_<=_n__}_ let D be set ; ::_thesis: ( D in meet {(dom (F . 0))} implies D in meet { (dom (F . k)) where k is Element of NAT : k <= n } ) assume D in meet {(dom (F . 0))} ; ::_thesis: D in meet { (dom (F . k)) where k is Element of NAT : k <= n } then A5: D in dom (F . 0) by SETFAM_1:10; A6: for E being set st E in { (dom (F . k)) where k is Element of NAT : k <= n } holds D in E proof let E be set ; ::_thesis: ( E in { (dom (F . k)) where k is Element of NAT : k <= n } implies D in E ) assume E in { (dom (F . k)) where k is Element of NAT : k <= n } ; ::_thesis: D in E then ex k1 being Element of NAT st ( E = dom (F . k1) & k1 <= n ) ; hence D in E by A2, A5, MESFUNC8:def_2; ::_thesis: verum end; dom (F . 0) in { (dom (F . k)) where k is Element of NAT : k <= n } ; hence D in meet { (dom (F . k)) where k is Element of NAT : k <= n } by A6, SETFAM_1:def_1; ::_thesis: verum end; then meet {(dom (F . 0))} c= meet { (dom (F . k)) where k is Element of NAT : k <= n } by TARSKI:def_3; then meet { (dom (F . k)) where k is Element of NAT : k <= n } = meet {(dom (F . 0))} by A4, XBOOLE_0:def_10; then dom ((Partial_Sums F) . n) = meet {(dom (F . 0))} by A1, Th28; hence dom ((Partial_Sums F) . n) = dom (F . 0) by SETFAM_1:10; ::_thesis: verum end; theorem Th30: :: MESFUNC9:30 for X being non empty set for F being Functional_Sequence of X,ExtREAL st ( for n being Nat holds F . n is nonnegative ) holds F is additive proof let X be non empty set ; ::_thesis: for F being Functional_Sequence of X,ExtREAL st ( for n being Nat holds F . n is nonnegative ) holds F is additive let F be Functional_Sequence of X,ExtREAL; ::_thesis: ( ( for n being Nat holds F . n is nonnegative ) implies F is additive ) assume A1: for n being Nat holds F . n is nonnegative ; ::_thesis: F is additive let n, m be Nat; :: according to MESFUNC9:def_5 ::_thesis: ( n <> m implies for x being set holds ( not x in (dom (F . n)) /\ (dom (F . m)) or (F . n) . x <> +infty or (F . m) . x <> -infty ) ) assume n <> m ; ::_thesis: for x being set holds ( not x in (dom (F . n)) /\ (dom (F . m)) or (F . n) . x <> +infty or (F . m) . x <> -infty ) F . m is nonnegative by A1; hence for x being set holds ( not x in (dom (F . n)) /\ (dom (F . m)) or (F . n) . x <> +infty or (F . m) . x <> -infty ) by SUPINF_2:51; ::_thesis: verum end; theorem Th31: :: MESFUNC9:31 for X being non empty set for F, G being Functional_Sequence of X,ExtREAL for D being set st F is additive & ( for n being Nat holds G . n = (F . n) | D ) holds G is additive proof let X be non empty set ; ::_thesis: for F, G being Functional_Sequence of X,ExtREAL for D being set st F is additive & ( for n being Nat holds G . n = (F . n) | D ) holds G is additive let F, G be Functional_Sequence of X,ExtREAL; ::_thesis: for D being set st F is additive & ( for n being Nat holds G . n = (F . n) | D ) holds G is additive let D be set ; ::_thesis: ( F is additive & ( for n being Nat holds G . n = (F . n) | D ) implies G is additive ) assume that A1: F is additive and A2: for n being Nat holds G . n = (F . n) | D ; ::_thesis: G is additive let n, m be Nat; :: according to MESFUNC9:def_5 ::_thesis: ( n <> m implies for x being set holds ( not x in (dom (G . n)) /\ (dom (G . m)) or (G . n) . x <> +infty or (G . m) . x <> -infty ) ) A3: G . m = (F . m) | D by A2; then A4: dom (G . m) c= dom (F . m) by RELAT_1:60; assume n <> m ; ::_thesis: for x being set holds ( not x in (dom (G . n)) /\ (dom (G . m)) or (G . n) . x <> +infty or (G . m) . x <> -infty ) let x be set ; ::_thesis: ( not x in (dom (G . n)) /\ (dom (G . m)) or (G . n) . x <> +infty or (G . m) . x <> -infty ) assume A5: x in (dom (G . n)) /\ (dom (G . m)) ; ::_thesis: ( (G . n) . x <> +infty or (G . m) . x <> -infty ) then A6: x in dom (G . m) by XBOOLE_0:def_4; A7: G . n = (F . n) | D by A2; then dom (G . n) c= dom (F . n) by RELAT_1:60; then (dom (G . n)) /\ (dom (G . m)) c= (dom (F . n)) /\ (dom (F . m)) by A4, XBOOLE_1:27; then A8: ( (F . n) . x <> +infty or (F . m) . x <> -infty ) by A1, A5, Def5; x in dom (G . n) by A5, XBOOLE_0:def_4; hence ( (G . n) . x <> +infty or (G . m) . x <> -infty ) by A7, A3, A8, A6, FUNCT_1:47; ::_thesis: verum end; theorem Th32: :: MESFUNC9:32 for X being non empty set for F being Functional_Sequence of X,ExtREAL for n being Nat for x being Element of X for D being set st F is additive & F is with_the_same_dom & D c= dom (F . 0) & x in D holds (Partial_Sums (F # x)) . n = ((Partial_Sums F) # x) . n proof let X be non empty set ; ::_thesis: for F being Functional_Sequence of X,ExtREAL for n being Nat for x being Element of X for D being set st F is additive & F is with_the_same_dom & D c= dom (F . 0) & x in D holds (Partial_Sums (F # x)) . n = ((Partial_Sums F) # x) . n let F be Functional_Sequence of X,ExtREAL; ::_thesis: for n being Nat for x being Element of X for D being set st F is additive & F is with_the_same_dom & D c= dom (F . 0) & x in D holds (Partial_Sums (F # x)) . n = ((Partial_Sums F) # x) . n let n be Nat; ::_thesis: for x being Element of X for D being set st F is additive & F is with_the_same_dom & D c= dom (F . 0) & x in D holds (Partial_Sums (F # x)) . n = ((Partial_Sums F) # x) . n let x be Element of X; ::_thesis: for D being set st F is additive & F is with_the_same_dom & D c= dom (F . 0) & x in D holds (Partial_Sums (F # x)) . n = ((Partial_Sums F) # x) . n let D be set ; ::_thesis: ( F is additive & F is with_the_same_dom & D c= dom (F . 0) & x in D implies (Partial_Sums (F # x)) . n = ((Partial_Sums F) # x) . n ) set PF = Partial_Sums F; set PFx = Partial_Sums (F # x); assume that A1: F is additive and A2: F is with_the_same_dom and A3: D c= dom (F . 0) and A4: x in D ; ::_thesis: (Partial_Sums (F # x)) . n = ((Partial_Sums F) # x) . n defpred S1[ Nat] means (Partial_Sums (F # x)) . $1 = ((Partial_Sums F) # x) . $1; A5: for k being Nat st S1[k] holds S1[k + 1] proof let k be Nat; ::_thesis: ( S1[k] implies S1[k + 1] ) assume A6: S1[k] ; ::_thesis: S1[k + 1] (Partial_Sums (F # x)) . (k + 1) = ((Partial_Sums (F # x)) . k) + ((F # x) . (k + 1)) by Def1; then (Partial_Sums (F # x)) . (k + 1) = (((Partial_Sums F) # x) . k) + ((F . (k + 1)) . x) by A6, MESFUNC5:def_13; then A7: (Partial_Sums (F # x)) . (k + 1) = (((Partial_Sums F) . k) . x) + ((F . (k + 1)) . x) by MESFUNC5:def_13; A8: (Partial_Sums F) . (k + 1) = ((Partial_Sums F) . k) + (F . (k + 1)) by Def4; D c= dom ((Partial_Sums F) . (k + 1)) by A1, A2, A3, Th29; then (Partial_Sums (F # x)) . (k + 1) = ((Partial_Sums F) . (k + 1)) . x by A4, A8, A7, MESFUNC1:def_3; hence S1[k + 1] by MESFUNC5:def_13; ::_thesis: verum end; (Partial_Sums (F # x)) . 0 = (F # x) . 0 by Def1; then (Partial_Sums (F # x)) . 0 = (F . 0) . x by MESFUNC5:def_13; then (Partial_Sums (F # x)) . 0 = ((Partial_Sums F) . 0) . x by Def4; then A9: S1[ 0 ] by MESFUNC5:def_13; for k being Nat holds S1[k] from NAT_1:sch_2(A9, A5); hence (Partial_Sums (F # x)) . n = ((Partial_Sums F) # x) . n ; ::_thesis: verum end; theorem Th33: :: MESFUNC9:33 for X being non empty set for F being Functional_Sequence of X,ExtREAL for x being Element of X for D being set st F is additive & F is with_the_same_dom & D c= dom (F . 0) & x in D holds ( ( Partial_Sums (F # x) is convergent_to_finite_number implies (Partial_Sums F) # x is convergent_to_finite_number ) & ( (Partial_Sums F) # x is convergent_to_finite_number implies Partial_Sums (F # x) is convergent_to_finite_number ) & ( Partial_Sums (F # x) is convergent_to_+infty implies (Partial_Sums F) # x is convergent_to_+infty ) & ( (Partial_Sums F) # x is convergent_to_+infty implies Partial_Sums (F # x) is convergent_to_+infty ) & ( Partial_Sums (F # x) is convergent_to_-infty implies (Partial_Sums F) # x is convergent_to_-infty ) & ( (Partial_Sums F) # x is convergent_to_-infty implies Partial_Sums (F # x) is convergent_to_-infty ) & ( Partial_Sums (F # x) is convergent implies (Partial_Sums F) # x is convergent ) & ( (Partial_Sums F) # x is convergent implies Partial_Sums (F # x) is convergent ) ) proof let X be non empty set ; ::_thesis: for F being Functional_Sequence of X,ExtREAL for x being Element of X for D being set st F is additive & F is with_the_same_dom & D c= dom (F . 0) & x in D holds ( ( Partial_Sums (F # x) is convergent_to_finite_number implies (Partial_Sums F) # x is convergent_to_finite_number ) & ( (Partial_Sums F) # x is convergent_to_finite_number implies Partial_Sums (F # x) is convergent_to_finite_number ) & ( Partial_Sums (F # x) is convergent_to_+infty implies (Partial_Sums F) # x is convergent_to_+infty ) & ( (Partial_Sums F) # x is convergent_to_+infty implies Partial_Sums (F # x) is convergent_to_+infty ) & ( Partial_Sums (F # x) is convergent_to_-infty implies (Partial_Sums F) # x is convergent_to_-infty ) & ( (Partial_Sums F) # x is convergent_to_-infty implies Partial_Sums (F # x) is convergent_to_-infty ) & ( Partial_Sums (F # x) is convergent implies (Partial_Sums F) # x is convergent ) & ( (Partial_Sums F) # x is convergent implies Partial_Sums (F # x) is convergent ) ) let F be Functional_Sequence of X,ExtREAL; ::_thesis: for x being Element of X for D being set st F is additive & F is with_the_same_dom & D c= dom (F . 0) & x in D holds ( ( Partial_Sums (F # x) is convergent_to_finite_number implies (Partial_Sums F) # x is convergent_to_finite_number ) & ( (Partial_Sums F) # x is convergent_to_finite_number implies Partial_Sums (F # x) is convergent_to_finite_number ) & ( Partial_Sums (F # x) is convergent_to_+infty implies (Partial_Sums F) # x is convergent_to_+infty ) & ( (Partial_Sums F) # x is convergent_to_+infty implies Partial_Sums (F # x) is convergent_to_+infty ) & ( Partial_Sums (F # x) is convergent_to_-infty implies (Partial_Sums F) # x is convergent_to_-infty ) & ( (Partial_Sums F) # x is convergent_to_-infty implies Partial_Sums (F # x) is convergent_to_-infty ) & ( Partial_Sums (F # x) is convergent implies (Partial_Sums F) # x is convergent ) & ( (Partial_Sums F) # x is convergent implies Partial_Sums (F # x) is convergent ) ) let x be Element of X; ::_thesis: for D being set st F is additive & F is with_the_same_dom & D c= dom (F . 0) & x in D holds ( ( Partial_Sums (F # x) is convergent_to_finite_number implies (Partial_Sums F) # x is convergent_to_finite_number ) & ( (Partial_Sums F) # x is convergent_to_finite_number implies Partial_Sums (F # x) is convergent_to_finite_number ) & ( Partial_Sums (F # x) is convergent_to_+infty implies (Partial_Sums F) # x is convergent_to_+infty ) & ( (Partial_Sums F) # x is convergent_to_+infty implies Partial_Sums (F # x) is convergent_to_+infty ) & ( Partial_Sums (F # x) is convergent_to_-infty implies (Partial_Sums F) # x is convergent_to_-infty ) & ( (Partial_Sums F) # x is convergent_to_-infty implies Partial_Sums (F # x) is convergent_to_-infty ) & ( Partial_Sums (F # x) is convergent implies (Partial_Sums F) # x is convergent ) & ( (Partial_Sums F) # x is convergent implies Partial_Sums (F # x) is convergent ) ) let D be set ; ::_thesis: ( F is additive & F is with_the_same_dom & D c= dom (F . 0) & x in D implies ( ( Partial_Sums (F # x) is convergent_to_finite_number implies (Partial_Sums F) # x is convergent_to_finite_number ) & ( (Partial_Sums F) # x is convergent_to_finite_number implies Partial_Sums (F # x) is convergent_to_finite_number ) & ( Partial_Sums (F # x) is convergent_to_+infty implies (Partial_Sums F) # x is convergent_to_+infty ) & ( (Partial_Sums F) # x is convergent_to_+infty implies Partial_Sums (F # x) is convergent_to_+infty ) & ( Partial_Sums (F # x) is convergent_to_-infty implies (Partial_Sums F) # x is convergent_to_-infty ) & ( (Partial_Sums F) # x is convergent_to_-infty implies Partial_Sums (F # x) is convergent_to_-infty ) & ( Partial_Sums (F # x) is convergent implies (Partial_Sums F) # x is convergent ) & ( (Partial_Sums F) # x is convergent implies Partial_Sums (F # x) is convergent ) ) ) set PFx = Partial_Sums (F # x); set PF = Partial_Sums F; assume that A1: F is additive and A2: F is with_the_same_dom and A3: D c= dom (F . 0) and A4: x in D ; ::_thesis: ( ( Partial_Sums (F # x) is convergent_to_finite_number implies (Partial_Sums F) # x is convergent_to_finite_number ) & ( (Partial_Sums F) # x is convergent_to_finite_number implies Partial_Sums (F # x) is convergent_to_finite_number ) & ( Partial_Sums (F # x) is convergent_to_+infty implies (Partial_Sums F) # x is convergent_to_+infty ) & ( (Partial_Sums F) # x is convergent_to_+infty implies Partial_Sums (F # x) is convergent_to_+infty ) & ( Partial_Sums (F # x) is convergent_to_-infty implies (Partial_Sums F) # x is convergent_to_-infty ) & ( (Partial_Sums F) # x is convergent_to_-infty implies Partial_Sums (F # x) is convergent_to_-infty ) & ( Partial_Sums (F # x) is convergent implies (Partial_Sums F) # x is convergent ) & ( (Partial_Sums F) # x is convergent implies Partial_Sums (F # x) is convergent ) ) hereby ::_thesis: ( ( (Partial_Sums F) # x is convergent_to_finite_number implies Partial_Sums (F # x) is convergent_to_finite_number ) & ( Partial_Sums (F # x) is convergent_to_+infty implies (Partial_Sums F) # x is convergent_to_+infty ) & ( (Partial_Sums F) # x is convergent_to_+infty implies Partial_Sums (F # x) is convergent_to_+infty ) & ( Partial_Sums (F # x) is convergent_to_-infty implies (Partial_Sums F) # x is convergent_to_-infty ) & ( (Partial_Sums F) # x is convergent_to_-infty implies Partial_Sums (F # x) is convergent_to_-infty ) & ( Partial_Sums (F # x) is convergent implies (Partial_Sums F) # x is convergent ) & ( (Partial_Sums F) # x is convergent implies Partial_Sums (F # x) is convergent ) ) assume Partial_Sums (F # x) is convergent_to_finite_number ; ::_thesis: (Partial_Sums F) # x is convergent_to_finite_number then consider g being real number such that A6: for p being real number st 0 < p holds ex n being Nat st for m being Nat st n <= m holds |.(((Partial_Sums (F # x)) . m) - (R_EAL g)).| < p by MESFUNC5:def_8; now__::_thesis:_for_p_being_real_number_st_0_<_p_holds_ ex_n_being_Nat_st_ for_m_being_Nat_st_n_<=_m_holds_ |.((((Partial_Sums_F)_#_x)_._m)_-_(R_EAL_g)).|_<_p let p be real number ; ::_thesis: ( 0 < p implies ex n being Nat st for m being Nat st n <= m holds |.((((Partial_Sums F) # x) . m) - (R_EAL g)).| < p ) assume 0 < p ; ::_thesis: ex n being Nat st for m being Nat st n <= m holds |.((((Partial_Sums F) # x) . m) - (R_EAL g)).| < p then consider n being Nat such that A7: for m being Nat st n <= m holds |.(((Partial_Sums (F # x)) . m) - (R_EAL g)).| < p by A6; take n = n; ::_thesis: for m being Nat st n <= m holds |.((((Partial_Sums F) # x) . m) - (R_EAL g)).| < p let m be Nat; ::_thesis: ( n <= m implies |.((((Partial_Sums F) # x) . m) - (R_EAL g)).| < p ) assume A8: n <= m ; ::_thesis: |.((((Partial_Sums F) # x) . m) - (R_EAL g)).| < p (Partial_Sums (F # x)) . m = ((Partial_Sums F) # x) . m by A1, A2, A3, A4, Th32; hence |.((((Partial_Sums F) # x) . m) - (R_EAL g)).| < p by A7, A8; ::_thesis: verum end; hence (Partial_Sums F) # x is convergent_to_finite_number by MESFUNC5:def_8; ::_thesis: verum end; hereby ::_thesis: ( ( Partial_Sums (F # x) is convergent_to_+infty implies (Partial_Sums F) # x is convergent_to_+infty ) & ( (Partial_Sums F) # x is convergent_to_+infty implies Partial_Sums (F # x) is convergent_to_+infty ) & ( Partial_Sums (F # x) is convergent_to_-infty implies (Partial_Sums F) # x is convergent_to_-infty ) & ( (Partial_Sums F) # x is convergent_to_-infty implies Partial_Sums (F # x) is convergent_to_-infty ) & ( Partial_Sums (F # x) is convergent implies (Partial_Sums F) # x is convergent ) & ( (Partial_Sums F) # x is convergent implies Partial_Sums (F # x) is convergent ) ) assume (Partial_Sums F) # x is convergent_to_finite_number ; ::_thesis: Partial_Sums (F # x) is convergent_to_finite_number then consider g being real number such that A10: for p being real number st 0 < p holds ex n being Nat st for m being Nat st n <= m holds |.((((Partial_Sums F) # x) . m) - (R_EAL g)).| < p by MESFUNC5:def_8; now__::_thesis:_for_p_being_real_number_st_0_<_p_holds_ ex_n_being_Nat_st_ for_m_being_Nat_st_n_<=_m_holds_ |.(((Partial_Sums_(F_#_x))_._m)_-_(R_EAL_g)).|_<_p let p be real number ; ::_thesis: ( 0 < p implies ex n being Nat st for m being Nat st n <= m holds |.(((Partial_Sums (F # x)) . m) - (R_EAL g)).| < p ) assume 0 < p ; ::_thesis: ex n being Nat st for m being Nat st n <= m holds |.(((Partial_Sums (F # x)) . m) - (R_EAL g)).| < p then consider n being Nat such that A11: for m being Nat st n <= m holds |.((((Partial_Sums F) # x) . m) - (R_EAL g)).| < p by A10; take n = n; ::_thesis: for m being Nat st n <= m holds |.(((Partial_Sums (F # x)) . m) - (R_EAL g)).| < p let m be Nat; ::_thesis: ( n <= m implies |.(((Partial_Sums (F # x)) . m) - (R_EAL g)).| < p ) assume A12: n <= m ; ::_thesis: |.(((Partial_Sums (F # x)) . m) - (R_EAL g)).| < p (Partial_Sums (F # x)) . m = ((Partial_Sums F) # x) . m by A1, A2, A3, A4, Th32; hence |.(((Partial_Sums (F # x)) . m) - (R_EAL g)).| < p by A11, A12; ::_thesis: verum end; hence Partial_Sums (F # x) is convergent_to_finite_number by MESFUNC5:def_8; ::_thesis: verum end; hereby ::_thesis: ( ( (Partial_Sums F) # x is convergent_to_+infty implies Partial_Sums (F # x) is convergent_to_+infty ) & ( Partial_Sums (F # x) is convergent_to_-infty implies (Partial_Sums F) # x is convergent_to_-infty ) & ( (Partial_Sums F) # x is convergent_to_-infty implies Partial_Sums (F # x) is convergent_to_-infty ) & ( Partial_Sums (F # x) is convergent implies (Partial_Sums F) # x is convergent ) & ( (Partial_Sums F) # x is convergent implies Partial_Sums (F # x) is convergent ) ) assume A14: Partial_Sums (F # x) is convergent_to_+infty ; ::_thesis: (Partial_Sums F) # x is convergent_to_+infty now__::_thesis:_for_r_being_real_number_st_0_<_r_holds_ ex_n_being_Nat_st_ for_m_being_Nat_st_n_<=_m_holds_ r_<=_((Partial_Sums_F)_#_x)_._m let r be real number ; ::_thesis: ( 0 < r implies ex n being Nat st for m being Nat st n <= m holds r <= ((Partial_Sums F) # x) . m ) assume 0 < r ; ::_thesis: ex n being Nat st for m being Nat st n <= m holds r <= ((Partial_Sums F) # x) . m then consider n being Nat such that A15: for m being Nat st n <= m holds r <= (Partial_Sums (F # x)) . m by A14, MESFUNC5:def_9; take n = n; ::_thesis: for m being Nat st n <= m holds r <= ((Partial_Sums F) # x) . m let m be Nat; ::_thesis: ( n <= m implies r <= ((Partial_Sums F) # x) . m ) assume n <= m ; ::_thesis: r <= ((Partial_Sums F) # x) . m then r <= (Partial_Sums (F # x)) . m by A15; hence r <= ((Partial_Sums F) # x) . m by A1, A2, A3, A4, Th32; ::_thesis: verum end; hence (Partial_Sums F) # x is convergent_to_+infty by MESFUNC5:def_9; ::_thesis: verum end; hereby ::_thesis: ( ( Partial_Sums (F # x) is convergent_to_-infty implies (Partial_Sums F) # x is convergent_to_-infty ) & ( (Partial_Sums F) # x is convergent_to_-infty implies Partial_Sums (F # x) is convergent_to_-infty ) & ( Partial_Sums (F # x) is convergent implies (Partial_Sums F) # x is convergent ) & ( (Partial_Sums F) # x is convergent implies Partial_Sums (F # x) is convergent ) ) assume A17: (Partial_Sums F) # x is convergent_to_+infty ; ::_thesis: Partial_Sums (F # x) is convergent_to_+infty now__::_thesis:_for_r_being_real_number_st_0_<_r_holds_ ex_n_being_Nat_st_ for_m_being_Nat_st_n_<=_m_holds_ r_<=_(Partial_Sums_(F_#_x))_._m let r be real number ; ::_thesis: ( 0 < r implies ex n being Nat st for m being Nat st n <= m holds r <= (Partial_Sums (F # x)) . m ) assume 0 < r ; ::_thesis: ex n being Nat st for m being Nat st n <= m holds r <= (Partial_Sums (F # x)) . m then consider n being Nat such that A18: for m being Nat st n <= m holds r <= ((Partial_Sums F) # x) . m by A17, MESFUNC5:def_9; take n = n; ::_thesis: for m being Nat st n <= m holds r <= (Partial_Sums (F # x)) . m let m be Nat; ::_thesis: ( n <= m implies r <= (Partial_Sums (F # x)) . m ) assume n <= m ; ::_thesis: r <= (Partial_Sums (F # x)) . m then r <= ((Partial_Sums F) # x) . m by A18; hence r <= (Partial_Sums (F # x)) . m by A1, A2, A3, A4, Th32; ::_thesis: verum end; hence Partial_Sums (F # x) is convergent_to_+infty by MESFUNC5:def_9; ::_thesis: verum end; hereby ::_thesis: ( ( (Partial_Sums F) # x is convergent_to_-infty implies Partial_Sums (F # x) is convergent_to_-infty ) & ( Partial_Sums (F # x) is convergent implies (Partial_Sums F) # x is convergent ) & ( (Partial_Sums F) # x is convergent implies Partial_Sums (F # x) is convergent ) ) assume A20: Partial_Sums (F # x) is convergent_to_-infty ; ::_thesis: (Partial_Sums F) # x is convergent_to_-infty now__::_thesis:_for_r_being_real_number_st_r_<_0_holds_ ex_n_being_Nat_st_ for_m_being_Nat_st_n_<=_m_holds_ ((Partial_Sums_F)_#_x)_._m_<=_r let r be real number ; ::_thesis: ( r < 0 implies ex n being Nat st for m being Nat st n <= m holds ((Partial_Sums F) # x) . m <= r ) assume r < 0 ; ::_thesis: ex n being Nat st for m being Nat st n <= m holds ((Partial_Sums F) # x) . m <= r then consider n being Nat such that A21: for m being Nat st n <= m holds (Partial_Sums (F # x)) . m <= r by A20, MESFUNC5:def_10; take n = n; ::_thesis: for m being Nat st n <= m holds ((Partial_Sums F) # x) . m <= r let m be Nat; ::_thesis: ( n <= m implies ((Partial_Sums F) # x) . m <= r ) assume n <= m ; ::_thesis: ((Partial_Sums F) # x) . m <= r then (Partial_Sums (F # x)) . m <= r by A21; hence ((Partial_Sums F) # x) . m <= r by A1, A2, A3, A4, Th32; ::_thesis: verum end; hence (Partial_Sums F) # x is convergent_to_-infty by MESFUNC5:def_10; ::_thesis: verum end; hereby ::_thesis: ( Partial_Sums (F # x) is convergent iff (Partial_Sums F) # x is convergent ) assume A23: (Partial_Sums F) # x is convergent_to_-infty ; ::_thesis: Partial_Sums (F # x) is convergent_to_-infty now__::_thesis:_for_r_being_real_number_st_r_<_0_holds_ ex_n_being_Nat_st_ for_m_being_Nat_st_n_<=_m_holds_ (Partial_Sums_(F_#_x))_._m_<=_r let r be real number ; ::_thesis: ( r < 0 implies ex n being Nat st for m being Nat st n <= m holds (Partial_Sums (F # x)) . m <= r ) assume r < 0 ; ::_thesis: ex n being Nat st for m being Nat st n <= m holds (Partial_Sums (F # x)) . m <= r then consider n being Nat such that A24: for m being Nat st n <= m holds ((Partial_Sums F) # x) . m <= r by A23, MESFUNC5:def_10; take n = n; ::_thesis: for m being Nat st n <= m holds (Partial_Sums (F # x)) . m <= r let m be Nat; ::_thesis: ( n <= m implies (Partial_Sums (F # x)) . m <= r ) assume n <= m ; ::_thesis: (Partial_Sums (F # x)) . m <= r then ((Partial_Sums F) # x) . m <= r by A24; hence (Partial_Sums (F # x)) . m <= r by A1, A2, A3, A4, Th32; ::_thesis: verum end; hence Partial_Sums (F # x) is convergent_to_-infty by MESFUNC5:def_10; ::_thesis: verum end; hereby ::_thesis: ( (Partial_Sums F) # x is convergent implies Partial_Sums (F # x) is convergent ) assume A25: Partial_Sums (F # x) is convergent ; ::_thesis: (Partial_Sums F) # x is convergent percases ( Partial_Sums (F # x) is convergent_to_+infty or Partial_Sums (F # x) is convergent_to_-infty or Partial_Sums (F # x) is convergent_to_finite_number ) by A25, MESFUNC5:def_11; suppose Partial_Sums (F # x) is convergent_to_+infty ; ::_thesis: (Partial_Sums F) # x is convergent hence (Partial_Sums F) # x is convergent by A13, MESFUNC5:def_11; ::_thesis: verum end; suppose Partial_Sums (F # x) is convergent_to_-infty ; ::_thesis: (Partial_Sums F) # x is convergent hence (Partial_Sums F) # x is convergent by A19, MESFUNC5:def_11; ::_thesis: verum end; suppose Partial_Sums (F # x) is convergent_to_finite_number ; ::_thesis: (Partial_Sums F) # x is convergent hence (Partial_Sums F) # x is convergent by A5, MESFUNC5:def_11; ::_thesis: verum end; end; end; assume A26: (Partial_Sums F) # x is convergent ; ::_thesis: Partial_Sums (F # x) is convergent percases ( (Partial_Sums F) # x is convergent_to_+infty or (Partial_Sums F) # x is convergent_to_-infty or (Partial_Sums F) # x is convergent_to_finite_number ) by A26, MESFUNC5:def_11; suppose (Partial_Sums F) # x is convergent_to_+infty ; ::_thesis: Partial_Sums (F # x) is convergent hence Partial_Sums (F # x) is convergent by A16, MESFUNC5:def_11; ::_thesis: verum end; suppose (Partial_Sums F) # x is convergent_to_-infty ; ::_thesis: Partial_Sums (F # x) is convergent hence Partial_Sums (F # x) is convergent by A22, MESFUNC5:def_11; ::_thesis: verum end; suppose (Partial_Sums F) # x is convergent_to_finite_number ; ::_thesis: Partial_Sums (F # x) is convergent hence Partial_Sums (F # x) is convergent by A9, MESFUNC5:def_11; ::_thesis: verum end; end; end; theorem Th34: :: MESFUNC9:34 for X being non empty set for F being Functional_Sequence of X,ExtREAL for f being PartFunc of X,ExtREAL for x being Element of X st F is additive & F is with_the_same_dom & dom f c= dom (F . 0) & x in dom f & F # x is summable & f . x = Sum (F # x) holds f . x = lim ((Partial_Sums F) # x) proof let X be non empty set ; ::_thesis: for F being Functional_Sequence of X,ExtREAL for f being PartFunc of X,ExtREAL for x being Element of X st F is additive & F is with_the_same_dom & dom f c= dom (F . 0) & x in dom f & F # x is summable & f . x = Sum (F # x) holds f . x = lim ((Partial_Sums F) # x) let F be Functional_Sequence of X,ExtREAL; ::_thesis: for f being PartFunc of X,ExtREAL for x being Element of X st F is additive & F is with_the_same_dom & dom f c= dom (F . 0) & x in dom f & F # x is summable & f . x = Sum (F # x) holds f . x = lim ((Partial_Sums F) # x) let f be PartFunc of X,ExtREAL; ::_thesis: for x being Element of X st F is additive & F is with_the_same_dom & dom f c= dom (F . 0) & x in dom f & F # x is summable & f . x = Sum (F # x) holds f . x = lim ((Partial_Sums F) # x) let x be Element of X; ::_thesis: ( F is additive & F is with_the_same_dom & dom f c= dom (F . 0) & x in dom f & F # x is summable & f . x = Sum (F # x) implies f . x = lim ((Partial_Sums F) # x) ) set PF = Partial_Sums F; assume that A1: F is additive and A2: F is with_the_same_dom and A3: dom f c= dom (F . 0) and A4: x in dom f and A5: F # x is summable and A6: f . x = Sum (F # x) ; ::_thesis: f . x = lim ((Partial_Sums F) # x) set PFx = Partial_Sums (F # x); Partial_Sums (F # x) is convergent by A5, Def2; then A7: (Partial_Sums F) # x is convergent by A1, A2, A3, A4, Th33; percases ( ex g being real number st ( lim ((Partial_Sums F) # x) = g & ( for p being real number st 0 < p holds ex n being Nat st for m being Nat st n <= m holds |.((((Partial_Sums F) # x) . m) - (lim ((Partial_Sums F) # x))).| < p ) & (Partial_Sums F) # x is convergent_to_finite_number ) or ( lim ((Partial_Sums F) # x) = +infty & (Partial_Sums F) # x is convergent_to_+infty ) or ( lim ((Partial_Sums F) # x) = -infty & (Partial_Sums F) # x is convergent_to_-infty ) ) by A7, MESFUNC5:def_12; supposeA8: ex g being real number st ( lim ((Partial_Sums F) # x) = g & ( for p being real number st 0 < p holds ex n being Nat st for m being Nat st n <= m holds |.((((Partial_Sums F) # x) . m) - (lim ((Partial_Sums F) # x))).| < p ) & (Partial_Sums F) # x is convergent_to_finite_number ) ; ::_thesis: f . x = lim ((Partial_Sums F) # x) then A9: Partial_Sums (F # x) is convergent_to_finite_number by A1, A2, A3, A4, Th33; then A10: not Partial_Sums (F # x) is convergent_to_+infty by MESFUNC5:50; A11: not Partial_Sums (F # x) is convergent_to_-infty by A9, MESFUNC5:51; Partial_Sums (F # x) is convergent by A9, MESFUNC5:def_11; then A12: ex g being real number st ( f . x = g & ( for p being real number st 0 < p holds ex n being Nat st for m being Nat st n <= m holds |.(((Partial_Sums (F # x)) . m) - (f . x)).| < p ) & Partial_Sums (F # x) is convergent_to_finite_number ) by A6, A10, A11, MESFUNC5:def_12; now__::_thesis:_for_p_being_real_number_st_0_<_p_holds_ ex_n_being_Nat_st_ for_m_being_Nat_st_n_<=_m_holds_ |.((((Partial_Sums_F)_#_x)_._m)_-_(f_._x)).|_<_p let p be real number ; ::_thesis: ( 0 < p implies ex n being Nat st for m being Nat st n <= m holds |.((((Partial_Sums F) # x) . m) - (f . x)).| < p ) assume 0 < p ; ::_thesis: ex n being Nat st for m being Nat st n <= m holds |.((((Partial_Sums F) # x) . m) - (f . x)).| < p then consider n being Nat such that A13: for m being Nat st n <= m holds |.(((Partial_Sums (F # x)) . m) - (f . x)).| < p by A12; take n = n; ::_thesis: for m being Nat st n <= m holds |.((((Partial_Sums F) # x) . m) - (f . x)).| < p let m be Nat; ::_thesis: ( n <= m implies |.((((Partial_Sums F) # x) . m) - (f . x)).| < p ) assume A14: n <= m ; ::_thesis: |.((((Partial_Sums F) # x) . m) - (f . x)).| < p (Partial_Sums (F # x)) . m = ((Partial_Sums F) # x) . m by A1, A2, A3, A4, Th32; hence |.((((Partial_Sums F) # x) . m) - (f . x)).| < p by A13, A14; ::_thesis: verum end; hence f . x = lim ((Partial_Sums F) # x) by A7, A8, A12, MESFUNC5:def_12; ::_thesis: verum end; supposeA15: ( lim ((Partial_Sums F) # x) = +infty & (Partial_Sums F) # x is convergent_to_+infty ) ; ::_thesis: f . x = lim ((Partial_Sums F) # x) then A16: Partial_Sums (F # x) is convergent_to_+infty by A1, A2, A3, A4, Th33; then Partial_Sums (F # x) is convergent by MESFUNC5:def_11; hence f . x = lim ((Partial_Sums F) # x) by A6, A15, A16, MESFUNC5:def_12; ::_thesis: verum end; supposeA17: ( lim ((Partial_Sums F) # x) = -infty & (Partial_Sums F) # x is convergent_to_-infty ) ; ::_thesis: f . x = lim ((Partial_Sums F) # x) then A18: Partial_Sums (F # x) is convergent_to_-infty by A1, A2, A3, A4, Th33; then Partial_Sums (F # x) is convergent by MESFUNC5:def_11; hence f . x = lim ((Partial_Sums F) # x) by A6, A17, A18, MESFUNC5:def_12; ::_thesis: verum end; end; end; theorem Th35: :: MESFUNC9:35 for X being non empty set for S being SigmaField of X for F being Functional_Sequence of X,ExtREAL for n being Nat st ( for m being Nat holds F . m is_simple_func_in S ) holds ( F is additive & (Partial_Sums F) . n is_simple_func_in S ) proof let X be non empty set ; ::_thesis: for S being SigmaField of X for F being Functional_Sequence of X,ExtREAL for n being Nat st ( for m being Nat holds F . m is_simple_func_in S ) holds ( F is additive & (Partial_Sums F) . n is_simple_func_in S ) let S be SigmaField of X; ::_thesis: for F being Functional_Sequence of X,ExtREAL for n being Nat st ( for m being Nat holds F . m is_simple_func_in S ) holds ( F is additive & (Partial_Sums F) . n is_simple_func_in S ) let F be Functional_Sequence of X,ExtREAL; ::_thesis: for n being Nat st ( for m being Nat holds F . m is_simple_func_in S ) holds ( F is additive & (Partial_Sums F) . n is_simple_func_in S ) let n be Nat; ::_thesis: ( ( for m being Nat holds F . m is_simple_func_in S ) implies ( F is additive & (Partial_Sums F) . n is_simple_func_in S ) ) defpred S1[ Nat] means (Partial_Sums F) . $1 is_simple_func_in S; assume A1: for m being Nat holds F . m is_simple_func_in S ; ::_thesis: ( F is additive & (Partial_Sums F) . n is_simple_func_in S ) hereby :: according to MESFUNC9:def_5 ::_thesis: (Partial_Sums F) . n is_simple_func_in S let n, m be Nat; ::_thesis: ( n <> m implies for x being set holds ( not x in (dom (F . n)) /\ (dom (F . m)) or (F . n) . x <> +infty or (F . m) . x <> -infty ) ) assume n <> m ; ::_thesis: for x being set holds ( not x in (dom (F . n)) /\ (dom (F . m)) or (F . n) . x <> +infty or (F . m) . x <> -infty ) F . n is_simple_func_in S by A1; then F . n is V120() by MESFUNC5:14; hence for x being set holds ( not x in (dom (F . n)) /\ (dom (F . m)) or (F . n) . x <> +infty or (F . m) . x <> -infty ) by MESFUNC5:def_6; ::_thesis: verum end; A2: for k being Nat st S1[k] holds S1[k + 1] proof let k be Nat; ::_thesis: ( S1[k] implies S1[k + 1] ) assume A3: S1[k] ; ::_thesis: S1[k + 1] F . (k + 1) is_simple_func_in S by A1; then ((Partial_Sums F) . k) + (F . (k + 1)) is_simple_func_in S by A3, MESFUNC5:38; hence S1[k + 1] by Def4; ::_thesis: verum end; (Partial_Sums F) . 0 = F . 0 by Def4; then A4: S1[ 0 ] by A1; for k being Nat holds S1[k] from NAT_1:sch_2(A4, A2); hence (Partial_Sums F) . n is_simple_func_in S ; ::_thesis: verum end; theorem Th36: :: MESFUNC9:36 for X being non empty set for F being Functional_Sequence of X,ExtREAL for n being Nat st ( for m being Nat holds F . m is nonnegative ) holds (Partial_Sums F) . n is nonnegative proof let X be non empty set ; ::_thesis: for F being Functional_Sequence of X,ExtREAL for n being Nat st ( for m being Nat holds F . m is nonnegative ) holds (Partial_Sums F) . n is nonnegative let F be Functional_Sequence of X,ExtREAL; ::_thesis: for n being Nat st ( for m being Nat holds F . m is nonnegative ) holds (Partial_Sums F) . n is nonnegative let n be Nat; ::_thesis: ( ( for m being Nat holds F . m is nonnegative ) implies (Partial_Sums F) . n is nonnegative ) defpred S1[ Nat] means (Partial_Sums F) . $1 is nonnegative ; assume A1: for m being Nat holds F . m is nonnegative ; ::_thesis: (Partial_Sums F) . n is nonnegative A2: now__::_thesis:_for_k_being_Nat_st_S1[k]_holds_ S1[k_+_1] let k be Nat; ::_thesis: ( S1[k] implies S1[k + 1] ) assume A3: S1[k] ; ::_thesis: S1[k + 1] A4: (Partial_Sums F) . (k + 1) = ((Partial_Sums F) . k) + (F . (k + 1)) by Def4; F . (k + 1) is nonnegative by A1; hence S1[k + 1] by A3, A4, MESFUNC5:22; ::_thesis: verum end; (Partial_Sums F) . 0 = F . 0 by Def4; then A5: S1[ 0 ] by A1; for k being Nat holds S1[k] from NAT_1:sch_2(A5, A2); hence (Partial_Sums F) . n is nonnegative ; ::_thesis: verum end; theorem Th37: :: MESFUNC9:37 for X being non empty set for F being Functional_Sequence of X,ExtREAL for n, m being Nat for x being Element of X st F is with_the_same_dom & x in dom (F . 0) & ( for k being Nat holds F . k is nonnegative ) & n <= m holds ((Partial_Sums F) . n) . x <= ((Partial_Sums F) . m) . x proof let X be non empty set ; ::_thesis: for F being Functional_Sequence of X,ExtREAL for n, m being Nat for x being Element of X st F is with_the_same_dom & x in dom (F . 0) & ( for k being Nat holds F . k is nonnegative ) & n <= m holds ((Partial_Sums F) . n) . x <= ((Partial_Sums F) . m) . x let F be Functional_Sequence of X,ExtREAL; ::_thesis: for n, m being Nat for x being Element of X st F is with_the_same_dom & x in dom (F . 0) & ( for k being Nat holds F . k is nonnegative ) & n <= m holds ((Partial_Sums F) . n) . x <= ((Partial_Sums F) . m) . x let n, m be Nat; ::_thesis: for x being Element of X st F is with_the_same_dom & x in dom (F . 0) & ( for k being Nat holds F . k is nonnegative ) & n <= m holds ((Partial_Sums F) . n) . x <= ((Partial_Sums F) . m) . x let x be Element of X; ::_thesis: ( F is with_the_same_dom & x in dom (F . 0) & ( for k being Nat holds F . k is nonnegative ) & n <= m implies ((Partial_Sums F) . n) . x <= ((Partial_Sums F) . m) . x ) assume A1: F is with_the_same_dom ; ::_thesis: ( not x in dom (F . 0) or ex k being Nat st not F . k is nonnegative or not n <= m or ((Partial_Sums F) . n) . x <= ((Partial_Sums F) . m) . x ) set PF = Partial_Sums F; assume A2: x in dom (F . 0) ; ::_thesis: ( ex k being Nat st not F . k is nonnegative or not n <= m or ((Partial_Sums F) . n) . x <= ((Partial_Sums F) . m) . x ) defpred S1[ Nat] means ((Partial_Sums F) . n) . x <= ((Partial_Sums F) . $1) . x; assume A3: for m being Nat holds F . m is nonnegative ; ::_thesis: ( not n <= m or ((Partial_Sums F) . n) . x <= ((Partial_Sums F) . m) . x ) A4: for k being Nat holds ((Partial_Sums F) . k) . x <= ((Partial_Sums F) . (k + 1)) . x proof let k be Nat; ::_thesis: ((Partial_Sums F) . k) . x <= ((Partial_Sums F) . (k + 1)) . x A5: (Partial_Sums F) . (k + 1) = ((Partial_Sums F) . k) + (F . (k + 1)) by Def4; F . (k + 1) is nonnegative by A3; then A6: 0. <= (F . (k + 1)) . x by SUPINF_2:39; dom ((Partial_Sums F) . (k + 1)) = dom (F . 0) by A1, A3, Th29, Th30; then ((Partial_Sums F) . (k + 1)) . x = (((Partial_Sums F) . k) . x) + ((F . (k + 1)) . x) by A2, A5, MESFUNC1:def_3; hence ((Partial_Sums F) . k) . x <= ((Partial_Sums F) . (k + 1)) . x by A6, XXREAL_3:39; ::_thesis: verum end; A7: for k being Nat st k >= n & ( for l being Nat st l >= n & l < k holds S1[l] ) holds S1[k] proof let k be Nat; ::_thesis: ( k >= n & ( for l being Nat st l >= n & l < k holds S1[l] ) implies S1[k] ) assume that A8: k >= n and A9: for l being Nat st l >= n & l < k holds S1[l] ; ::_thesis: S1[k] now__::_thesis:_(_k_>_n_implies_S1[k]_) A10: now__::_thesis:_(_k_>_n_+_1_implies_S1[k]_) assume A11: k > n + 1 ; ::_thesis: S1[k] then reconsider l = k - 1 as Element of NAT by NAT_1:20; k < k + 1 by NAT_1:13; then A12: k > l by XREAL_1:19; k = l + 1 ; then A13: ((Partial_Sums F) . l) . x <= ((Partial_Sums F) . k) . x by A4; l >= n by A11, XREAL_1:19; then ((Partial_Sums F) . n) . x <= ((Partial_Sums F) . l) . x by A9, A12; hence S1[k] by A13, XXREAL_0:2; ::_thesis: verum end; assume k > n ; ::_thesis: S1[k] then k >= n + 1 by NAT_1:13; then ( k = n + 1 or k > n + 1 ) by XXREAL_0:1; hence S1[k] by A4, A10; ::_thesis: verum end; hence S1[k] by A8, XXREAL_0:1; ::_thesis: verum end; A14: for k being Nat st k >= n holds S1[k] from NAT_1:sch_9(A7); assume n <= m ; ::_thesis: ((Partial_Sums F) . n) . x <= ((Partial_Sums F) . m) . x hence ((Partial_Sums F) . n) . x <= ((Partial_Sums F) . m) . x by A14; ::_thesis: verum end; theorem Th38: :: MESFUNC9:38 for X being non empty set for F being Functional_Sequence of X,ExtREAL for x being Element of X st F is with_the_same_dom & x in dom (F . 0) & ( for m being Nat holds F . m is nonnegative ) holds ( (Partial_Sums F) # x is non-decreasing & (Partial_Sums F) # x is convergent ) proof let X be non empty set ; ::_thesis: for F being Functional_Sequence of X,ExtREAL for x being Element of X st F is with_the_same_dom & x in dom (F . 0) & ( for m being Nat holds F . m is nonnegative ) holds ( (Partial_Sums F) # x is non-decreasing & (Partial_Sums F) # x is convergent ) let F be Functional_Sequence of X,ExtREAL; ::_thesis: for x being Element of X st F is with_the_same_dom & x in dom (F . 0) & ( for m being Nat holds F . m is nonnegative ) holds ( (Partial_Sums F) # x is non-decreasing & (Partial_Sums F) # x is convergent ) let x be Element of X; ::_thesis: ( F is with_the_same_dom & x in dom (F . 0) & ( for m being Nat holds F . m is nonnegative ) implies ( (Partial_Sums F) # x is non-decreasing & (Partial_Sums F) # x is convergent ) ) assume A1: F is with_the_same_dom ; ::_thesis: ( not x in dom (F . 0) or ex m being Nat st not F . m is nonnegative or ( (Partial_Sums F) # x is non-decreasing & (Partial_Sums F) # x is convergent ) ) assume A2: x in dom (F . 0) ; ::_thesis: ( ex m being Nat st not F . m is nonnegative or ( (Partial_Sums F) # x is non-decreasing & (Partial_Sums F) # x is convergent ) ) assume A3: for m being Nat holds F . m is nonnegative ; ::_thesis: ( (Partial_Sums F) # x is non-decreasing & (Partial_Sums F) # x is convergent ) for n, m being Element of NAT st m <= n holds ((Partial_Sums F) # x) . m <= ((Partial_Sums F) # x) . n proof let n, m be Element of NAT ; ::_thesis: ( m <= n implies ((Partial_Sums F) # x) . m <= ((Partial_Sums F) # x) . n ) assume m <= n ; ::_thesis: ((Partial_Sums F) # x) . m <= ((Partial_Sums F) # x) . n then ((Partial_Sums F) . m) . x <= ((Partial_Sums F) . n) . x by A1, A2, A3, Th37; then ((Partial_Sums F) # x) . m <= ((Partial_Sums F) . n) . x by MESFUNC5:def_13; hence ((Partial_Sums F) # x) . m <= ((Partial_Sums F) # x) . n by MESFUNC5:def_13; ::_thesis: verum end; hence (Partial_Sums F) # x is non-decreasing by RINFSUP2:7; ::_thesis: (Partial_Sums F) # x is convergent hence (Partial_Sums F) # x is convergent by RINFSUP2:37; ::_thesis: verum end; theorem Th39: :: MESFUNC9:39 for X being non empty set for F being Functional_Sequence of X,ExtREAL for n being Nat st ( for m being Nat holds F . m is V119() ) holds (Partial_Sums F) . n is V119() proof let X be non empty set ; ::_thesis: for F being Functional_Sequence of X,ExtREAL for n being Nat st ( for m being Nat holds F . m is V119() ) holds (Partial_Sums F) . n is V119() let F be Functional_Sequence of X,ExtREAL; ::_thesis: for n being Nat st ( for m being Nat holds F . m is V119() ) holds (Partial_Sums F) . n is V119() let n be Nat; ::_thesis: ( ( for m being Nat holds F . m is V119() ) implies (Partial_Sums F) . n is V119() ) defpred S1[ Nat] means (Partial_Sums F) . $1 is V119(); assume A1: for m being Nat holds F . m is V119() ; ::_thesis: (Partial_Sums F) . n is V119() A2: for k being Nat st S1[k] holds S1[k + 1] proof let k be Nat; ::_thesis: ( S1[k] implies S1[k + 1] ) assume A3: S1[k] ; ::_thesis: S1[k + 1] A4: F . (k + 1) is V119() by A1; (Partial_Sums F) . (k + 1) = ((Partial_Sums F) . k) + (F . (k + 1)) by Def4; hence S1[k + 1] by A3, A4, Th3; ::_thesis: verum end; (Partial_Sums F) . 0 = F . 0 by Def4; then A5: S1[ 0 ] by A1; for k being Nat holds S1[k] from NAT_1:sch_2(A5, A2); hence (Partial_Sums F) . n is V119() ; ::_thesis: verum end; theorem :: MESFUNC9:40 for X being non empty set for F being Functional_Sequence of X,ExtREAL for n being Nat st ( for m being Nat holds F . m is V120() ) holds (Partial_Sums F) . n is V120() proof let X be non empty set ; ::_thesis: for F being Functional_Sequence of X,ExtREAL for n being Nat st ( for m being Nat holds F . m is V120() ) holds (Partial_Sums F) . n is V120() let F be Functional_Sequence of X,ExtREAL; ::_thesis: for n being Nat st ( for m being Nat holds F . m is V120() ) holds (Partial_Sums F) . n is V120() let n be Nat; ::_thesis: ( ( for m being Nat holds F . m is V120() ) implies (Partial_Sums F) . n is V120() ) defpred S1[ Nat] means (Partial_Sums F) . $1 is V120(); assume A1: for m being Nat holds F . m is V120() ; ::_thesis: (Partial_Sums F) . n is V120() A2: for k being Nat st S1[k] holds S1[k + 1] proof let k be Nat; ::_thesis: ( S1[k] implies S1[k + 1] ) assume A3: S1[k] ; ::_thesis: S1[k + 1] A4: F . (k + 1) is V120() by A1; (Partial_Sums F) . (k + 1) = ((Partial_Sums F) . k) + (F . (k + 1)) by Def4; hence S1[k + 1] by A3, A4, Th4; ::_thesis: verum end; (Partial_Sums F) . 0 = F . 0 by Def4; then A5: S1[ 0 ] by A1; for k being Nat holds S1[k] from NAT_1:sch_2(A5, A2); hence (Partial_Sums F) . n is V120() ; ::_thesis: verum end; theorem Th41: :: MESFUNC9:41 for X being non empty set for S being SigmaField of X for E being Element of S for F being Functional_Sequence of X,ExtREAL for m being Nat st ( for n being Nat holds ( F . n is_measurable_on E & F . n is V119() ) ) holds (Partial_Sums F) . m is_measurable_on E proof let X be non empty set ; ::_thesis: for S being SigmaField of X for E being Element of S for F being Functional_Sequence of X,ExtREAL for m being Nat st ( for n being Nat holds ( F . n is_measurable_on E & F . n is V119() ) ) holds (Partial_Sums F) . m is_measurable_on E let S be SigmaField of X; ::_thesis: for E being Element of S for F being Functional_Sequence of X,ExtREAL for m being Nat st ( for n being Nat holds ( F . n is_measurable_on E & F . n is V119() ) ) holds (Partial_Sums F) . m is_measurable_on E let E be Element of S; ::_thesis: for F being Functional_Sequence of X,ExtREAL for m being Nat st ( for n being Nat holds ( F . n is_measurable_on E & F . n is V119() ) ) holds (Partial_Sums F) . m is_measurable_on E let F be Functional_Sequence of X,ExtREAL; ::_thesis: for m being Nat st ( for n being Nat holds ( F . n is_measurable_on E & F . n is V119() ) ) holds (Partial_Sums F) . m is_measurable_on E let m be Nat; ::_thesis: ( ( for n being Nat holds ( F . n is_measurable_on E & F . n is V119() ) ) implies (Partial_Sums F) . m is_measurable_on E ) set PF = Partial_Sums F; defpred S1[ Nat] means (Partial_Sums F) . $1 is_measurable_on E; assume A1: for n being Nat holds ( F . n is_measurable_on E & F . n is V119() ) ; ::_thesis: (Partial_Sums F) . m is_measurable_on E A2: for k being Nat st S1[k] holds S1[k + 1] proof let k be Nat; ::_thesis: ( S1[k] implies S1[k + 1] ) assume A3: S1[k] ; ::_thesis: S1[k + 1] A4: F . (k + 1) is_measurable_on E by A1; A5: F . (k + 1) is V119() by A1; (Partial_Sums F) . k is V119() by A1, Th39; then ((Partial_Sums F) . k) + (F . (k + 1)) is_measurable_on E by A3, A4, A5, MESFUNC5:31; hence S1[k + 1] by Def4; ::_thesis: verum end; (Partial_Sums F) . 0 = F . 0 by Def4; then A6: S1[ 0 ] by A1; for k being Nat holds S1[k] from NAT_1:sch_2(A6, A2); hence (Partial_Sums F) . m is_measurable_on E ; ::_thesis: verum end; theorem Th42: :: MESFUNC9:42 for X being non empty set for F, G being Functional_Sequence of X,ExtREAL for n being Nat for x being Element of X st F is additive & F is with_the_same_dom & G is additive & G is with_the_same_dom & x in (dom (F . 0)) /\ (dom (G . 0)) & ( for k being Nat for y being Element of X st y in (dom (F . 0)) /\ (dom (G . 0)) holds (F . k) . y <= (G . k) . y ) holds ((Partial_Sums F) . n) . x <= ((Partial_Sums G) . n) . x proof let X be non empty set ; ::_thesis: for F, G being Functional_Sequence of X,ExtREAL for n being Nat for x being Element of X st F is additive & F is with_the_same_dom & G is additive & G is with_the_same_dom & x in (dom (F . 0)) /\ (dom (G . 0)) & ( for k being Nat for y being Element of X st y in (dom (F . 0)) /\ (dom (G . 0)) holds (F . k) . y <= (G . k) . y ) holds ((Partial_Sums F) . n) . x <= ((Partial_Sums G) . n) . x let F, G be Functional_Sequence of X,ExtREAL; ::_thesis: for n being Nat for x being Element of X st F is additive & F is with_the_same_dom & G is additive & G is with_the_same_dom & x in (dom (F . 0)) /\ (dom (G . 0)) & ( for k being Nat for y being Element of X st y in (dom (F . 0)) /\ (dom (G . 0)) holds (F . k) . y <= (G . k) . y ) holds ((Partial_Sums F) . n) . x <= ((Partial_Sums G) . n) . x let n be Nat; ::_thesis: for x being Element of X st F is additive & F is with_the_same_dom & G is additive & G is with_the_same_dom & x in (dom (F . 0)) /\ (dom (G . 0)) & ( for k being Nat for y being Element of X st y in (dom (F . 0)) /\ (dom (G . 0)) holds (F . k) . y <= (G . k) . y ) holds ((Partial_Sums F) . n) . x <= ((Partial_Sums G) . n) . x let x be Element of X; ::_thesis: ( F is additive & F is with_the_same_dom & G is additive & G is with_the_same_dom & x in (dom (F . 0)) /\ (dom (G . 0)) & ( for k being Nat for y being Element of X st y in (dom (F . 0)) /\ (dom (G . 0)) holds (F . k) . y <= (G . k) . y ) implies ((Partial_Sums F) . n) . x <= ((Partial_Sums G) . n) . x ) assume that A1: F is additive and A2: F is with_the_same_dom and A3: G is additive and A4: G is with_the_same_dom and A5: x in (dom (F . 0)) /\ (dom (G . 0)) and A6: for k being Nat for y being Element of X st y in (dom (F . 0)) /\ (dom (G . 0)) holds (F . k) . y <= (G . k) . y ; ::_thesis: ((Partial_Sums F) . n) . x <= ((Partial_Sums G) . n) . x set PG = Partial_Sums G; set PF = Partial_Sums F; defpred S1[ Nat] means ((Partial_Sums F) . $1) . x <= ((Partial_Sums G) . $1) . x; A7: for k being Nat st S1[k] holds S1[k + 1] proof let k be Nat; ::_thesis: ( S1[k] implies S1[k + 1] ) assume A8: S1[k] ; ::_thesis: S1[k + 1] dom ((Partial_Sums F) . (k + 1)) = dom (F . 0) by A1, A2, Th29; then A9: x in dom ((Partial_Sums F) . (k + 1)) by A5, XBOOLE_0:def_4; dom ((Partial_Sums G) . (k + 1)) = dom (G . 0) by A3, A4, Th29; then A10: x in dom ((Partial_Sums G) . (k + 1)) by A5, XBOOLE_0:def_4; (Partial_Sums G) . (k + 1) = ((Partial_Sums G) . k) + (G . (k + 1)) by Def4; then A11: ((Partial_Sums G) . (k + 1)) . x = (((Partial_Sums G) . k) . x) + ((G . (k + 1)) . x) by A10, MESFUNC1:def_3; (Partial_Sums F) . (k + 1) = ((Partial_Sums F) . k) + (F . (k + 1)) by Def4; then A12: ((Partial_Sums F) . (k + 1)) . x = (((Partial_Sums F) . k) . x) + ((F . (k + 1)) . x) by A9, MESFUNC1:def_3; (F . (k + 1)) . x <= (G . (k + 1)) . x by A5, A6; hence S1[k + 1] by A8, A12, A11, XXREAL_3:36; ::_thesis: verum end; A13: (Partial_Sums G) . 0 = G . 0 by Def4; (Partial_Sums F) . 0 = F . 0 by Def4; then A14: S1[ 0 ] by A5, A6, A13; for k being Nat holds S1[k] from NAT_1:sch_2(A14, A7); hence ((Partial_Sums F) . n) . x <= ((Partial_Sums G) . n) . x ; ::_thesis: verum end; theorem Th43: :: MESFUNC9:43 for X being non empty set for F being Functional_Sequence of X,ExtREAL st F is additive & F is with_the_same_dom holds Partial_Sums F is with_the_same_dom proof let X be non empty set ; ::_thesis: for F being Functional_Sequence of X,ExtREAL st F is additive & F is with_the_same_dom holds Partial_Sums F is with_the_same_dom let F be Functional_Sequence of X,ExtREAL; ::_thesis: ( F is additive & F is with_the_same_dom implies Partial_Sums F is with_the_same_dom ) assume that A1: F is additive and A2: F is with_the_same_dom ; ::_thesis: Partial_Sums F is with_the_same_dom let n, m be Nat; :: according to MESFUNC8:def_2 ::_thesis: dom ((Partial_Sums F) . n) = dom ((Partial_Sums F) . m) dom ((Partial_Sums F) . n) = dom (F . 0) by A1, A2, Th29; hence dom ((Partial_Sums F) . n) = dom ((Partial_Sums F) . m) by A1, A2, Th29; ::_thesis: verum end; theorem Th44: :: MESFUNC9:44 for X being non empty set for S being SigmaField of X for E being Element of S for F being Functional_Sequence of X,ExtREAL st dom (F . 0) = E & F is additive & F is with_the_same_dom & ( for n being Nat holds (Partial_Sums F) . n is_measurable_on E ) & ( for x being Element of X st x in E holds F # x is summable ) holds lim (Partial_Sums F) is_measurable_on E proof let X be non empty set ; ::_thesis: for S being SigmaField of X for E being Element of S for F being Functional_Sequence of X,ExtREAL st dom (F . 0) = E & F is additive & F is with_the_same_dom & ( for n being Nat holds (Partial_Sums F) . n is_measurable_on E ) & ( for x being Element of X st x in E holds F # x is summable ) holds lim (Partial_Sums F) is_measurable_on E let S be SigmaField of X; ::_thesis: for E being Element of S for F being Functional_Sequence of X,ExtREAL st dom (F . 0) = E & F is additive & F is with_the_same_dom & ( for n being Nat holds (Partial_Sums F) . n is_measurable_on E ) & ( for x being Element of X st x in E holds F # x is summable ) holds lim (Partial_Sums F) is_measurable_on E let E be Element of S; ::_thesis: for F being Functional_Sequence of X,ExtREAL st dom (F . 0) = E & F is additive & F is with_the_same_dom & ( for n being Nat holds (Partial_Sums F) . n is_measurable_on E ) & ( for x being Element of X st x in E holds F # x is summable ) holds lim (Partial_Sums F) is_measurable_on E let F be Functional_Sequence of X,ExtREAL; ::_thesis: ( dom (F . 0) = E & F is additive & F is with_the_same_dom & ( for n being Nat holds (Partial_Sums F) . n is_measurable_on E ) & ( for x being Element of X st x in E holds F # x is summable ) implies lim (Partial_Sums F) is_measurable_on E ) assume that A1: dom (F . 0) = E and A2: F is additive and A3: F is with_the_same_dom and A4: for n being Nat holds (Partial_Sums F) . n is_measurable_on E and A5: for x being Element of X st x in E holds F # x is summable ; ::_thesis: lim (Partial_Sums F) is_measurable_on E reconsider PF = Partial_Sums F as with_the_same_dom Functional_Sequence of X,ExtREAL by A2, A3, Th43; A6: now__::_thesis:_for_x_being_Element_of_X_st_x_in_E_holds_ PF_#_x_is_convergent let x be Element of X; ::_thesis: ( x in E implies PF # x is convergent ) assume A7: x in E ; ::_thesis: PF # x is convergent then F # x is summable by A5; then Partial_Sums (F # x) is convergent by Def2; hence PF # x is convergent by A1, A2, A3, A7, Th33; ::_thesis: verum end; dom ((Partial_Sums F) . 0) = E by A1, A2, A3, Th29; hence lim (Partial_Sums F) is_measurable_on E by A4, A6, MESFUNC8:25; ::_thesis: verum end; theorem :: MESFUNC9:45 for X being non empty set for S being SigmaField of X for M being sigma_Measure of S for F being Functional_Sequence of X,ExtREAL st ( for n being Nat holds F . n is_integrable_on M ) holds for m being Nat holds (Partial_Sums F) . m is_integrable_on M proof let X be non empty set ; ::_thesis: for S being SigmaField of X for M being sigma_Measure of S for F being Functional_Sequence of X,ExtREAL st ( for n being Nat holds F . n is_integrable_on M ) holds for m being Nat holds (Partial_Sums F) . m is_integrable_on M let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S for F being Functional_Sequence of X,ExtREAL st ( for n being Nat holds F . n is_integrable_on M ) holds for m being Nat holds (Partial_Sums F) . m is_integrable_on M let M be sigma_Measure of S; ::_thesis: for F being Functional_Sequence of X,ExtREAL st ( for n being Nat holds F . n is_integrable_on M ) holds for m being Nat holds (Partial_Sums F) . m is_integrable_on M let F be Functional_Sequence of X,ExtREAL; ::_thesis: ( ( for n being Nat holds F . n is_integrable_on M ) implies for m being Nat holds (Partial_Sums F) . m is_integrable_on M ) set PF = Partial_Sums F; defpred S1[ Nat] means (Partial_Sums F) . $1 is_integrable_on M; assume A1: for n being Nat holds F . n is_integrable_on M ; ::_thesis: for m being Nat holds (Partial_Sums F) . m is_integrable_on M A2: for k being Nat st S1[k] holds S1[k + 1] proof let k be Nat; ::_thesis: ( S1[k] implies S1[k + 1] ) assume A3: S1[k] ; ::_thesis: S1[k + 1] F . (k + 1) is_integrable_on M by A1; then ((Partial_Sums F) . k) + (F . (k + 1)) is_integrable_on M by A3, MESFUNC5:108; hence S1[k + 1] by Def4; ::_thesis: verum end; (Partial_Sums F) . 0 = F . 0 by Def4; then A4: S1[ 0 ] by A1; thus for m being Nat holds S1[m] from NAT_1:sch_2(A4, A2); ::_thesis: verum end; theorem Th46: :: MESFUNC9:46 for X being non empty set for S being SigmaField of X for M being sigma_Measure of S for E being Element of S for F being Functional_Sequence of X,ExtREAL for I being ExtREAL_sequence for m being Nat st E = dom (F . 0) & F is additive & F is with_the_same_dom & ( for n being Nat holds ( F . n is_measurable_on E & F . n is nonnegative & I . n = Integral (M,(F . n)) ) ) holds Integral (M,((Partial_Sums F) . m)) = (Partial_Sums I) . m proof let X be non empty set ; ::_thesis: for S being SigmaField of X for M being sigma_Measure of S for E being Element of S for F being Functional_Sequence of X,ExtREAL for I being ExtREAL_sequence for m being Nat st E = dom (F . 0) & F is additive & F is with_the_same_dom & ( for n being Nat holds ( F . n is_measurable_on E & F . n is nonnegative & I . n = Integral (M,(F . n)) ) ) holds Integral (M,((Partial_Sums F) . m)) = (Partial_Sums I) . m let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S for E being Element of S for F being Functional_Sequence of X,ExtREAL for I being ExtREAL_sequence for m being Nat st E = dom (F . 0) & F is additive & F is with_the_same_dom & ( for n being Nat holds ( F . n is_measurable_on E & F . n is nonnegative & I . n = Integral (M,(F . n)) ) ) holds Integral (M,((Partial_Sums F) . m)) = (Partial_Sums I) . m let M be sigma_Measure of S; ::_thesis: for E being Element of S for F being Functional_Sequence of X,ExtREAL for I being ExtREAL_sequence for m being Nat st E = dom (F . 0) & F is additive & F is with_the_same_dom & ( for n being Nat holds ( F . n is_measurable_on E & F . n is nonnegative & I . n = Integral (M,(F . n)) ) ) holds Integral (M,((Partial_Sums F) . m)) = (Partial_Sums I) . m let E be Element of S; ::_thesis: for F being Functional_Sequence of X,ExtREAL for I being ExtREAL_sequence for m being Nat st E = dom (F . 0) & F is additive & F is with_the_same_dom & ( for n being Nat holds ( F . n is_measurable_on E & F . n is nonnegative & I . n = Integral (M,(F . n)) ) ) holds Integral (M,((Partial_Sums F) . m)) = (Partial_Sums I) . m let F be Functional_Sequence of X,ExtREAL; ::_thesis: for I being ExtREAL_sequence for m being Nat st E = dom (F . 0) & F is additive & F is with_the_same_dom & ( for n being Nat holds ( F . n is_measurable_on E & F . n is nonnegative & I . n = Integral (M,(F . n)) ) ) holds Integral (M,((Partial_Sums F) . m)) = (Partial_Sums I) . m let I be ExtREAL_sequence; ::_thesis: for m being Nat st E = dom (F . 0) & F is additive & F is with_the_same_dom & ( for n being Nat holds ( F . n is_measurable_on E & F . n is nonnegative & I . n = Integral (M,(F . n)) ) ) holds Integral (M,((Partial_Sums F) . m)) = (Partial_Sums I) . m let m be Nat; ::_thesis: ( E = dom (F . 0) & F is additive & F is with_the_same_dom & ( for n being Nat holds ( F . n is_measurable_on E & F . n is nonnegative & I . n = Integral (M,(F . n)) ) ) implies Integral (M,((Partial_Sums F) . m)) = (Partial_Sums I) . m ) assume that A1: E = dom (F . 0) and A2: F is additive and A3: F is with_the_same_dom and A4: for n being Nat holds ( F . n is_measurable_on E & F . n is nonnegative & I . n = Integral (M,(F . n)) ) ; ::_thesis: Integral (M,((Partial_Sums F) . m)) = (Partial_Sums I) . m set PF = Partial_Sums F; A5: for n being Nat holds F . n is V119() by A4, MESFUNC5:12; thus Integral (M,((Partial_Sums F) . m)) = (Partial_Sums I) . m ::_thesis: verum proof set PI = Partial_Sums I; defpred S1[ Nat] means Integral (M,((Partial_Sums F) . $1)) = (Partial_Sums I) . $1; A6: for k being Nat st S1[k] holds S1[k + 1] proof let k be Nat; ::_thesis: ( S1[k] implies S1[k + 1] ) assume A7: S1[k] ; ::_thesis: S1[k + 1] A8: F . (k + 1) is_measurable_on E by A4; A9: dom (F . (k + 1)) = E by A1, A3, MESFUNC8:def_2; A10: (Partial_Sums F) . (k + 1) is_measurable_on E by A4, A5, Th41; A11: (Partial_Sums F) . (k + 1) is nonnegative by A4, Th36; A12: F . (k + 1) is nonnegative by A4; A13: (Partial_Sums F) . k is nonnegative by A4, Th36; A14: dom ((Partial_Sums F) . k) = E by A1, A2, A3, Th29; A15: (Partial_Sums F) . k is_measurable_on E by A4, A5, Th41; then consider D being Element of S such that A16: D = dom (((Partial_Sums F) . k) + (F . (k + 1))) and A17: integral+ (M,(((Partial_Sums F) . k) + (F . (k + 1)))) = (integral+ (M,(((Partial_Sums F) . k) | D))) + (integral+ (M,((F . (k + 1)) | D))) by A14, A9, A8, A13, A12, MESFUNC5:78; A18: D = E /\ E by A14, A9, A13, A12, A16, MESFUNC5:22; then A19: ((Partial_Sums F) . k) | D = (Partial_Sums F) . k by A14, RELAT_1:68; A20: (F . (k + 1)) | D = F . (k + 1) by A9, A18, RELAT_1:68; dom ((Partial_Sums F) . (k + 1)) = E by A1, A2, A3, Th29; then Integral (M,((Partial_Sums F) . (k + 1))) = integral+ (M,((Partial_Sums F) . (k + 1))) by A10, A11, MESFUNC5:88 .= (integral+ (M,(((Partial_Sums F) . k) | D))) + (integral+ (M,((F . (k + 1)) | D))) by A17, Def4 .= (Integral (M,((Partial_Sums F) . k))) + (integral+ (M,((F . (k + 1)) | D))) by A14, A15, A13, A19, MESFUNC5:88 .= (Integral (M,((Partial_Sums F) . k))) + (Integral (M,(F . (k + 1)))) by A9, A8, A12, A20, MESFUNC5:88 .= ((Partial_Sums I) . k) + (I . (k + 1)) by A4, A7 ; hence S1[k + 1] by Def1; ::_thesis: verum end; Integral (M,((Partial_Sums F) . 0)) = Integral (M,(F . 0)) by Def4; then Integral (M,((Partial_Sums F) . 0)) = I . 0 by A4; then A21: S1[ 0 ] by Def1; for k being Nat holds S1[k] from NAT_1:sch_2(A21, A6); hence Integral (M,((Partial_Sums F) . m)) = (Partial_Sums I) . m ; ::_thesis: verum end; end; begin Lm2: for X being non empty set for S being SigmaField of X for M being sigma_Measure of S for E being Element of S for F being Functional_Sequence of X,ExtREAL for f being PartFunc of X,ExtREAL st E c= dom f & f is_measurable_on E & E = {} & ( for n being Nat holds F . n is_simple_func_in S ) holds ex I being ExtREAL_sequence st ( ( for n being Nat holds I . n = Integral (M,((F . n) | E)) ) & I is summable & Integral (M,(f | E)) = Sum I ) proof let X be non empty set ; ::_thesis: for S being SigmaField of X for M being sigma_Measure of S for E being Element of S for F being Functional_Sequence of X,ExtREAL for f being PartFunc of X,ExtREAL st E c= dom f & f is_measurable_on E & E = {} & ( for n being Nat holds F . n is_simple_func_in S ) holds ex I being ExtREAL_sequence st ( ( for n being Nat holds I . n = Integral (M,((F . n) | E)) ) & I is summable & Integral (M,(f | E)) = Sum I ) let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S for E being Element of S for F being Functional_Sequence of X,ExtREAL for f being PartFunc of X,ExtREAL st E c= dom f & f is_measurable_on E & E = {} & ( for n being Nat holds F . n is_simple_func_in S ) holds ex I being ExtREAL_sequence st ( ( for n being Nat holds I . n = Integral (M,((F . n) | E)) ) & I is summable & Integral (M,(f | E)) = Sum I ) let M be sigma_Measure of S; ::_thesis: for E being Element of S for F being Functional_Sequence of X,ExtREAL for f being PartFunc of X,ExtREAL st E c= dom f & f is_measurable_on E & E = {} & ( for n being Nat holds F . n is_simple_func_in S ) holds ex I being ExtREAL_sequence st ( ( for n being Nat holds I . n = Integral (M,((F . n) | E)) ) & I is summable & Integral (M,(f | E)) = Sum I ) let E be Element of S; ::_thesis: for F being Functional_Sequence of X,ExtREAL for f being PartFunc of X,ExtREAL st E c= dom f & f is_measurable_on E & E = {} & ( for n being Nat holds F . n is_simple_func_in S ) holds ex I being ExtREAL_sequence st ( ( for n being Nat holds I . n = Integral (M,((F . n) | E)) ) & I is summable & Integral (M,(f | E)) = Sum I ) let F be Functional_Sequence of X,ExtREAL; ::_thesis: for f being PartFunc of X,ExtREAL st E c= dom f & f is_measurable_on E & E = {} & ( for n being Nat holds F . n is_simple_func_in S ) holds ex I being ExtREAL_sequence st ( ( for n being Nat holds I . n = Integral (M,((F . n) | E)) ) & I is summable & Integral (M,(f | E)) = Sum I ) let f be PartFunc of X,ExtREAL; ::_thesis: ( E c= dom f & f is_measurable_on E & E = {} & ( for n being Nat holds F . n is_simple_func_in S ) implies ex I being ExtREAL_sequence st ( ( for n being Nat holds I . n = Integral (M,((F . n) | E)) ) & I is summable & Integral (M,(f | E)) = Sum I ) ) assume that A1: E c= dom f and A2: f is_measurable_on E and A3: E = {} and A4: for n being Nat holds F . n is_simple_func_in S ; ::_thesis: ex I being ExtREAL_sequence st ( ( for n being Nat holds I . n = Integral (M,((F . n) | E)) ) & I is summable & Integral (M,(f | E)) = Sum I ) take I = NAT --> 0.; ::_thesis: ( ( for n being Nat holds I . n = Integral (M,((F . n) | E)) ) & I is summable & Integral (M,(f | E)) = Sum I ) A5: M . E = 0 by A3, VALUED_0:def_19; thus for n being Nat holds I . n = Integral (M,((F . n) | E)) ::_thesis: ( I is summable & Integral (M,(f | E)) = Sum I ) proof let n be Nat; ::_thesis: I . n = Integral (M,((F . n) | E)) reconsider m = n as Element of NAT by ORDINAL1:def_12; reconsider D = dom (F . m) as Element of S by A4, MESFUNC5:37; F . m is_measurable_on D by A4, MESFUNC2:34; then Integral (M,((F . m) | E)) = 0 by A5, MESFUNC5:94; hence I . n = Integral (M,((F . n) | E)) by FUNCOP_1:7; ::_thesis: verum end; defpred S1[ Element of NAT ] means (Partial_Sums I) . $1 = 0 ; A6: for k being Element of NAT st S1[k] holds S1[k + 1] proof let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] ) assume A7: S1[k] ; ::_thesis: S1[k + 1] A8: I . (k + 1) = 0 by FUNCOP_1:7; (Partial_Sums I) . (k + 1) = ((Partial_Sums I) . k) + (I . (k + 1)) by Def1; hence S1[k + 1] by A7, A8; ::_thesis: verum end; (Partial_Sums I) . 0 = I . 0 by Def1; then A9: S1[ 0 ] by FUNCOP_1:7; A10: for k being Element of NAT holds S1[k] from NAT_1:sch_1(A9, A6); A11: for n being Nat holds (Partial_Sums I) . n = 0 proof let n be Nat; ::_thesis: (Partial_Sums I) . n = 0 n is Element of NAT by ORDINAL1:def_12; hence (Partial_Sums I) . n = 0 by A10; ::_thesis: verum end; then A12: lim (Partial_Sums I) = 0 by MESFUNC5:52; Partial_Sums I is convergent_to_finite_number by A11, MESFUNC5:52; then Partial_Sums I is convergent by MESFUNC5:def_11; hence I is summable by Def2; ::_thesis: Integral (M,(f | E)) = Sum I A13: E = dom (f | E) by A1, RELAT_1:62; then E = (dom f) /\ E by RELAT_1:61; then Integral (M,((f | E) | E)) = 0 by A2, A5, A13, MESFUNC5:42, MESFUNC5:94; hence Integral (M,(f | E)) = Sum I by A12; ::_thesis: verum end; Lm3: for X being non empty set for S being SigmaField of X for M being sigma_Measure of S for E being Element of S for F being Functional_Sequence of X,ExtREAL for f being PartFunc of X,ExtREAL st E c= dom f & f is nonnegative & f is_measurable_on E & F is additive & E common_on_dom F & ( for n being Nat holds ( F . n is_simple_func_in S & F . n is nonnegative ) ) & ( for x being Element of X st x in E holds ( F # x is summable & f . x = Sum (F # x) ) ) holds ex I being ExtREAL_sequence st ( ( for n being Nat holds I . n = Integral (M,((F . n) | E)) ) & I is summable & Integral (M,(f | E)) = Sum I ) proof let X be non empty set ; ::_thesis: for S being SigmaField of X for M being sigma_Measure of S for E being Element of S for F being Functional_Sequence of X,ExtREAL for f being PartFunc of X,ExtREAL st E c= dom f & f is nonnegative & f is_measurable_on E & F is additive & E common_on_dom F & ( for n being Nat holds ( F . n is_simple_func_in S & F . n is nonnegative ) ) & ( for x being Element of X st x in E holds ( F # x is summable & f . x = Sum (F # x) ) ) holds ex I being ExtREAL_sequence st ( ( for n being Nat holds I . n = Integral (M,((F . n) | E)) ) & I is summable & Integral (M,(f | E)) = Sum I ) let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S for E being Element of S for F being Functional_Sequence of X,ExtREAL for f being PartFunc of X,ExtREAL st E c= dom f & f is nonnegative & f is_measurable_on E & F is additive & E common_on_dom F & ( for n being Nat holds ( F . n is_simple_func_in S & F . n is nonnegative ) ) & ( for x being Element of X st x in E holds ( F # x is summable & f . x = Sum (F # x) ) ) holds ex I being ExtREAL_sequence st ( ( for n being Nat holds I . n = Integral (M,((F . n) | E)) ) & I is summable & Integral (M,(f | E)) = Sum I ) let M be sigma_Measure of S; ::_thesis: for E being Element of S for F being Functional_Sequence of X,ExtREAL for f being PartFunc of X,ExtREAL st E c= dom f & f is nonnegative & f is_measurable_on E & F is additive & E common_on_dom F & ( for n being Nat holds ( F . n is_simple_func_in S & F . n is nonnegative ) ) & ( for x being Element of X st x in E holds ( F # x is summable & f . x = Sum (F # x) ) ) holds ex I being ExtREAL_sequence st ( ( for n being Nat holds I . n = Integral (M,((F . n) | E)) ) & I is summable & Integral (M,(f | E)) = Sum I ) let E be Element of S; ::_thesis: for F being Functional_Sequence of X,ExtREAL for f being PartFunc of X,ExtREAL st E c= dom f & f is nonnegative & f is_measurable_on E & F is additive & E common_on_dom F & ( for n being Nat holds ( F . n is_simple_func_in S & F . n is nonnegative ) ) & ( for x being Element of X st x in E holds ( F # x is summable & f . x = Sum (F # x) ) ) holds ex I being ExtREAL_sequence st ( ( for n being Nat holds I . n = Integral (M,((F . n) | E)) ) & I is summable & Integral (M,(f | E)) = Sum I ) let F be Functional_Sequence of X,ExtREAL; ::_thesis: for f being PartFunc of X,ExtREAL st E c= dom f & f is nonnegative & f is_measurable_on E & F is additive & E common_on_dom F & ( for n being Nat holds ( F . n is_simple_func_in S & F . n is nonnegative ) ) & ( for x being Element of X st x in E holds ( F # x is summable & f . x = Sum (F # x) ) ) holds ex I being ExtREAL_sequence st ( ( for n being Nat holds I . n = Integral (M,((F . n) | E)) ) & I is summable & Integral (M,(f | E)) = Sum I ) let f be PartFunc of X,ExtREAL; ::_thesis: ( E c= dom f & f is nonnegative & f is_measurable_on E & F is additive & E common_on_dom F & ( for n being Nat holds ( F . n is_simple_func_in S & F . n is nonnegative ) ) & ( for x being Element of X st x in E holds ( F # x is summable & f . x = Sum (F # x) ) ) implies ex I being ExtREAL_sequence st ( ( for n being Nat holds I . n = Integral (M,((F . n) | E)) ) & I is summable & Integral (M,(f | E)) = Sum I ) ) assume that A1: E c= dom f and A2: f is nonnegative and A3: f is_measurable_on E and A4: F is additive and A5: E common_on_dom F and A6: for n being Nat holds ( F . n is_simple_func_in S & F . n is nonnegative ) and A7: for x being Element of X st x in E holds ( F # x is summable & f . x = Sum (F # x) ) ; ::_thesis: ex I being ExtREAL_sequence st ( ( for n being Nat holds I . n = Integral (M,((F . n) | E)) ) & I is summable & Integral (M,(f | E)) = Sum I ) deffunc H1( Nat) -> Element of bool [:X,ExtREAL:] = (F . $1) | E; consider g1 being Functional_Sequence of X,ExtREAL such that A8: for n being Nat holds g1 . n = H1(n) from SEQFUNC:sch_1(); A9: for n being Nat holds ( (F . n) | E is_simple_func_in S & (F . n) | E is nonnegative & dom ((F . n) | E) = E ) proof let n be Nat; ::_thesis: ( (F . n) | E is_simple_func_in S & (F . n) | E is nonnegative & dom ((F . n) | E) = E ) reconsider n9 = n as Element of NAT by ORDINAL1:def_12; thus (F . n) | E is_simple_func_in S by A6, MESFUNC5:34; ::_thesis: ( (F . n) | E is nonnegative & dom ((F . n) | E) = E ) thus (F . n) | E is nonnegative by A6, MESFUNC5:15; ::_thesis: dom ((F . n) | E) = E E c= dom (F . n9) by A5, SEQFUNC:def_9; hence dom ((F . n) | E) = E by RELAT_1:62; ::_thesis: verum end; for n, m being Nat holds dom (g1 . n) = dom (g1 . m) proof let n, m be Nat; ::_thesis: dom (g1 . n) = dom (g1 . m) dom (g1 . m) = dom ((F . m) | E) by A8; then A10: dom (g1 . m) = E by A9; dom (g1 . n) = dom ((F . n) | E) by A8; hence dom (g1 . n) = dom (g1 . m) by A9, A10; ::_thesis: verum end; then A11: g1 is with_the_same_dom by MESFUNC8:def_2; set G = Partial_Sums g1; deffunc H2( Element of NAT ) -> Element of ExtREAL = Integral (M,(g1 . $1)); consider I being ExtREAL_sequence such that A12: for n being Element of NAT holds I . n = H2(n) from FUNCT_2:sch_4(); take I ; ::_thesis: ( ( for n being Nat holds I . n = Integral (M,((F . n) | E)) ) & I is summable & Integral (M,(f | E)) = Sum I ) A13: dom (f | E) = E by A1, RELAT_1:62; then (dom f) /\ E = E by RELAT_1:61; then A14: f | E is_measurable_on E by A3, MESFUNC5:42; set L = Partial_Sums I; A15: for n being Nat holds I . n = Integral (M,((F . n) | E)) proof let n be Nat; ::_thesis: I . n = Integral (M,((F . n) | E)) reconsider m = n as Element of NAT by ORDINAL1:def_12; I . m = Integral (M,(g1 . m)) by A12; hence I . n = Integral (M,((F . n) | E)) by A8; ::_thesis: verum end; A16: for k being Nat holds g1 . k is nonnegative proof let k be Nat; ::_thesis: g1 . k is nonnegative (F . k) | E is nonnegative by A9; hence g1 . k is nonnegative by A8; ::_thesis: verum end; A17: for n being Nat holds ( (Partial_Sums g1) . n is_simple_func_in S & (Partial_Sums g1) . n is nonnegative & dom ((Partial_Sums g1) . n) = E ) proof let n be Nat; ::_thesis: ( (Partial_Sums g1) . n is_simple_func_in S & (Partial_Sums g1) . n is nonnegative & dom ((Partial_Sums g1) . n) = E ) A18: for n being Nat holds g1 . n is_simple_func_in S proof let n be Nat; ::_thesis: g1 . n is_simple_func_in S (F . n) | E is_simple_func_in S by A9; hence g1 . n is_simple_func_in S by A8; ::_thesis: verum end; hence (Partial_Sums g1) . n is_simple_func_in S by Th35; ::_thesis: ( (Partial_Sums g1) . n is nonnegative & dom ((Partial_Sums g1) . n) = E ) thus (Partial_Sums g1) . n is nonnegative by A16, Th36; ::_thesis: dom ((Partial_Sums g1) . n) = E dom (g1 . 0) = dom ((F . 0) | E) by A8; then dom (g1 . 0) = E by A9; hence dom ((Partial_Sums g1) . n) = E by A11, A18, Th29, Th35; ::_thesis: verum end; (Partial_Sums g1) . 0 = g1 . 0 by Def4; then A19: (Partial_Sums g1) . 0 = (F . 0) | E by A8; A20: for n being Nat holds integral' (M,((Partial_Sums g1) . n)) = (Partial_Sums I) . n proof defpred S1[ Element of NAT ] means (Partial_Sums I) . $1 = integral' (M,((Partial_Sums g1) . $1)); let n be Nat; ::_thesis: integral' (M,((Partial_Sums g1) . n)) = (Partial_Sums I) . n A21: (Partial_Sums g1) . 0 is nonnegative by A17; A22: for k being Element of NAT st S1[k] holds S1[k + 1] proof let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] ) assume A23: S1[k] ; ::_thesis: S1[k + 1] (Partial_Sums I) . (k + 1) = ((Partial_Sums I) . k) + (I . (k + 1)) by Def1; then A24: (Partial_Sums I) . (k + 1) = (integral' (M,((Partial_Sums g1) . k))) + (Integral (M,((F . (k + 1)) | E))) by A15, A23; A25: (F . (k + 1)) | E is_simple_func_in S by A9; A26: dom ((F . (k + 1)) | E) = E by A9; A27: (Partial_Sums g1) . k is_simple_func_in S by A17; (Partial_Sums g1) . (k + 1) = ((Partial_Sums g1) . k) + (g1 . (k + 1)) by Def4; then A28: (Partial_Sums g1) . (k + 1) = ((Partial_Sums g1) . k) + ((F . (k + 1)) | E) by A8; A29: (F . (k + 1)) | E is nonnegative by A9; A30: (Partial_Sums g1) . k is nonnegative by A17; A31: dom ((Partial_Sums g1) . k) = E by A17; then E = (dom ((Partial_Sums g1) . k)) /\ (dom ((F . (k + 1)) | E)) by A26; then dom (((Partial_Sums g1) . k) + ((F . (k + 1)) | E)) = E by A25, A29, A27, A30, MESFUNC5:65; then A32: integral' (M,(((Partial_Sums g1) . k) + ((F . (k + 1)) | E))) = (integral' (M,(((Partial_Sums g1) . k) | E))) + (integral' (M,(((F . (k + 1)) | E) | E))) by A25, A29, A27, A30, MESFUNC5:65; ((Partial_Sums g1) . k) | E = (Partial_Sums g1) . k by A31, RELAT_1:68; hence S1[k + 1] by A28, A24, A25, A29, A32, MESFUNC5:89; ::_thesis: verum end; (Partial_Sums I) . 0 = I . 0 by Def1; then A33: (Partial_Sums I) . 0 = Integral (M,((Partial_Sums g1) . 0)) by A15, A19; (Partial_Sums g1) . 0 is_simple_func_in S by A17; then A34: S1[ 0 ] by A33, A21, MESFUNC5:89; A35: for k being Element of NAT holds S1[k] from NAT_1:sch_1(A34, A22); n is Element of NAT by ORDINAL1:def_12; hence integral' (M,((Partial_Sums g1) . n)) = (Partial_Sums I) . n by A35; ::_thesis: verum end; g1 . 0 = (F . 0) | E by A8; then A36: dom (g1 . 0) = E by A9; A37: for x being Element of X st x in dom (f | E) holds ( g1 # x is summable & (f | E) . x = Sum (g1 # x) ) proof let x be Element of X; ::_thesis: ( x in dom (f | E) implies ( g1 # x is summable & (f | E) . x = Sum (g1 # x) ) ) assume A38: x in dom (f | E) ; ::_thesis: ( g1 # x is summable & (f | E) . x = Sum (g1 # x) ) then A39: f . x = (f | E) . x by FUNCT_1:47; A40: for n being set st n in NAT holds (F # x) . n = (g1 # x) . n proof let n be set ; ::_thesis: ( n in NAT implies (F # x) . n = (g1 # x) . n ) assume n in NAT ; ::_thesis: (F # x) . n = (g1 # x) . n then reconsider n1 = n as Nat ; A41: (F # x) . n = (F . n1) . x by MESFUNC5:def_13; A42: dom ((F . n1) | E) = E by A9; (F . n1) | E = g1 . n1 by A8; then (g1 . n1) . x = (F . n1) . x by A13, A38, A42, FUNCT_1:47; hence (F # x) . n = (g1 # x) . n by A41, MESFUNC5:def_13; ::_thesis: verum end; A43: f . x = Sum (F # x) by A7, A13, A38; F # x is summable by A7, A13, A38; hence ( g1 # x is summable & (f | E) . x = Sum (g1 # x) ) by A43, A39, A40, FUNCT_2:12; ::_thesis: verum end; A44: f | E is nonnegative by A2, MESFUNC5:15; then consider F being Functional_Sequence of X,ExtREAL, K being ExtREAL_sequence such that A45: for n being Nat holds ( F . n is_simple_func_in S & dom (F . n) = dom (f | E) ) and A46: for n being Nat holds F . n is nonnegative and A47: for n, m being Nat st n <= m holds for x being Element of X st x in dom (f | E) holds (F . n) . x <= (F . m) . x and A48: for x being Element of X st x in dom (f | E) holds ( F # x is convergent & lim (F # x) = (f | E) . x ) and A49: for n being Nat holds K . n = integral' (M,(F . n)) and K is convergent and A50: integral+ (M,(f | E)) = lim K by A13, A14, MESFUNC5:def_15; A51: g1 is additive by A4, A8, Th31; A52: for x being Element of X st x in E holds ( F # x is convergent & (Partial_Sums g1) # x is convergent & lim (F # x) = lim ((Partial_Sums g1) # x) ) proof let x be Element of X; ::_thesis: ( x in E implies ( F # x is convergent & (Partial_Sums g1) # x is convergent & lim (F # x) = lim ((Partial_Sums g1) # x) ) ) assume A53: x in E ; ::_thesis: ( F # x is convergent & (Partial_Sums g1) # x is convergent & lim (F # x) = lim ((Partial_Sums g1) # x) ) hence F # x is convergent by A13, A48; ::_thesis: ( (Partial_Sums g1) # x is convergent & lim (F # x) = lim ((Partial_Sums g1) # x) ) g1 # x is summable by A13, A37, A53; then Partial_Sums (g1 # x) is convergent by Def2; hence (Partial_Sums g1) # x is convergent by A11, A51, A36, A53, Th33; ::_thesis: lim (F # x) = lim ((Partial_Sums g1) # x) A54: (f | E) . x = Sum (g1 # x) by A13, A37, A53; g1 # x is summable by A13, A37, A53; then lim ((Partial_Sums g1) # x) = (f | E) . x by A4, A13, A8, A11, A36, A53, A54, Th31, Th34; hence lim (F # x) = lim ((Partial_Sums g1) # x) by A13, A48, A53; ::_thesis: verum end; A55: for n, m being Nat st n <= m holds for x being Element of X st x in E holds ((Partial_Sums g1) . n) . x <= ((Partial_Sums g1) . m) . x by A11, A16, A36, Th37; then A56: Partial_Sums I is convergent by A13, A17, A20, A45, A46, A47, A49, A52, MESFUNC5:76; lim (Partial_Sums I) = integral+ (M,(f | E)) by A13, A17, A20, A45, A46, A47, A49, A50, A55, A52, MESFUNC5:76; hence ( ( for n being Nat holds I . n = Integral (M,((F . n) | E)) ) & I is summable & Integral (M,(f | E)) = Sum I ) by A13, A15, A14, A44, A56, Def2, MESFUNC5:88; ::_thesis: verum end; theorem :: MESFUNC9:47 for X being non empty set for S being SigmaField of X for M being sigma_Measure of S for E being Element of S for F being Functional_Sequence of X,ExtREAL for f being PartFunc of X,ExtREAL st E c= dom f & f is nonnegative & f is_measurable_on E & F is additive & ( for n being Nat holds ( F . n is_simple_func_in S & F . n is nonnegative & E c= dom (F . n) ) ) & ( for x being Element of X st x in E holds ( F # x is summable & f . x = Sum (F # x) ) ) holds ex I being ExtREAL_sequence st ( ( for n being Nat holds I . n = Integral (M,((F . n) | E)) ) & I is summable & Integral (M,(f | E)) = Sum I ) proof let X be non empty set ; ::_thesis: for S being SigmaField of X for M being sigma_Measure of S for E being Element of S for F being Functional_Sequence of X,ExtREAL for f being PartFunc of X,ExtREAL st E c= dom f & f is nonnegative & f is_measurable_on E & F is additive & ( for n being Nat holds ( F . n is_simple_func_in S & F . n is nonnegative & E c= dom (F . n) ) ) & ( for x being Element of X st x in E holds ( F # x is summable & f . x = Sum (F # x) ) ) holds ex I being ExtREAL_sequence st ( ( for n being Nat holds I . n = Integral (M,((F . n) | E)) ) & I is summable & Integral (M,(f | E)) = Sum I ) let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S for E being Element of S for F being Functional_Sequence of X,ExtREAL for f being PartFunc of X,ExtREAL st E c= dom f & f is nonnegative & f is_measurable_on E & F is additive & ( for n being Nat holds ( F . n is_simple_func_in S & F . n is nonnegative & E c= dom (F . n) ) ) & ( for x being Element of X st x in E holds ( F # x is summable & f . x = Sum (F # x) ) ) holds ex I being ExtREAL_sequence st ( ( for n being Nat holds I . n = Integral (M,((F . n) | E)) ) & I is summable & Integral (M,(f | E)) = Sum I ) let M be sigma_Measure of S; ::_thesis: for E being Element of S for F being Functional_Sequence of X,ExtREAL for f being PartFunc of X,ExtREAL st E c= dom f & f is nonnegative & f is_measurable_on E & F is additive & ( for n being Nat holds ( F . n is_simple_func_in S & F . n is nonnegative & E c= dom (F . n) ) ) & ( for x being Element of X st x in E holds ( F # x is summable & f . x = Sum (F # x) ) ) holds ex I being ExtREAL_sequence st ( ( for n being Nat holds I . n = Integral (M,((F . n) | E)) ) & I is summable & Integral (M,(f | E)) = Sum I ) let E be Element of S; ::_thesis: for F being Functional_Sequence of X,ExtREAL for f being PartFunc of X,ExtREAL st E c= dom f & f is nonnegative & f is_measurable_on E & F is additive & ( for n being Nat holds ( F . n is_simple_func_in S & F . n is nonnegative & E c= dom (F . n) ) ) & ( for x being Element of X st x in E holds ( F # x is summable & f . x = Sum (F # x) ) ) holds ex I being ExtREAL_sequence st ( ( for n being Nat holds I . n = Integral (M,((F . n) | E)) ) & I is summable & Integral (M,(f | E)) = Sum I ) let F be Functional_Sequence of X,ExtREAL; ::_thesis: for f being PartFunc of X,ExtREAL st E c= dom f & f is nonnegative & f is_measurable_on E & F is additive & ( for n being Nat holds ( F . n is_simple_func_in S & F . n is nonnegative & E c= dom (F . n) ) ) & ( for x being Element of X st x in E holds ( F # x is summable & f . x = Sum (F # x) ) ) holds ex I being ExtREAL_sequence st ( ( for n being Nat holds I . n = Integral (M,((F . n) | E)) ) & I is summable & Integral (M,(f | E)) = Sum I ) let f be PartFunc of X,ExtREAL; ::_thesis: ( E c= dom f & f is nonnegative & f is_measurable_on E & F is additive & ( for n being Nat holds ( F . n is_simple_func_in S & F . n is nonnegative & E c= dom (F . n) ) ) & ( for x being Element of X st x in E holds ( F # x is summable & f . x = Sum (F # x) ) ) implies ex I being ExtREAL_sequence st ( ( for n being Nat holds I . n = Integral (M,((F . n) | E)) ) & I is summable & Integral (M,(f | E)) = Sum I ) ) assume that A1: E c= dom f and A2: f is nonnegative and A3: f is_measurable_on E and A4: F is additive and A5: for n being Nat holds ( F . n is_simple_func_in S & F . n is nonnegative & E c= dom (F . n) ) and A6: for x being Element of X st x in E holds ( F # x is summable & f . x = Sum (F # x) ) ; ::_thesis: ex I being ExtREAL_sequence st ( ( for n being Nat holds I . n = Integral (M,((F . n) | E)) ) & I is summable & Integral (M,(f | E)) = Sum I ) percases ( E = {} or E <> {} ) ; suppose E = {} ; ::_thesis: ex I being ExtREAL_sequence st ( ( for n being Nat holds I . n = Integral (M,((F . n) | E)) ) & I is summable & Integral (M,(f | E)) = Sum I ) hence ex I being ExtREAL_sequence st ( ( for n being Nat holds I . n = Integral (M,((F . n) | E)) ) & I is summable & Integral (M,(f | E)) = Sum I ) by A1, A3, A5, Lm2; ::_thesis: verum end; supposeA7: E <> {} ; ::_thesis: ex I being ExtREAL_sequence st ( ( for n being Nat holds I . n = Integral (M,((F . n) | E)) ) & I is summable & Integral (M,(f | E)) = Sum I ) for n being Element of NAT holds ( F . n is_simple_func_in S & F . n is nonnegative & E c= dom (F . n) ) by A5; then E common_on_dom F by A7, SEQFUNC:def_9; hence ex I being ExtREAL_sequence st ( ( for n being Nat holds I . n = Integral (M,((F . n) | E)) ) & I is summable & Integral (M,(f | E)) = Sum I ) by A1, A2, A3, A4, A5, A6, Lm3; ::_thesis: verum end; end; end; theorem :: MESFUNC9:48 for X being non empty set for S being SigmaField of X for M being sigma_Measure of S for E being Element of S for f being PartFunc of X,ExtREAL st E c= dom f & f is nonnegative & f is_measurable_on E holds ex g being Functional_Sequence of X,ExtREAL st ( g is additive & ( for n being Nat holds ( g . n is_simple_func_in S & g . n is nonnegative & g . n is_measurable_on E ) ) & ( for x being Element of X st x in E holds ( g # x is summable & f . x = Sum (g # x) ) ) & ex I being ExtREAL_sequence st ( ( for n being Nat holds I . n = Integral (M,((g . n) | E)) ) & I is summable & Integral (M,(f | E)) = Sum I ) ) proof let X be non empty set ; ::_thesis: for S being SigmaField of X for M being sigma_Measure of S for E being Element of S for f being PartFunc of X,ExtREAL st E c= dom f & f is nonnegative & f is_measurable_on E holds ex g being Functional_Sequence of X,ExtREAL st ( g is additive & ( for n being Nat holds ( g . n is_simple_func_in S & g . n is nonnegative & g . n is_measurable_on E ) ) & ( for x being Element of X st x in E holds ( g # x is summable & f . x = Sum (g # x) ) ) & ex I being ExtREAL_sequence st ( ( for n being Nat holds I . n = Integral (M,((g . n) | E)) ) & I is summable & Integral (M,(f | E)) = Sum I ) ) let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S for E being Element of S for f being PartFunc of X,ExtREAL st E c= dom f & f is nonnegative & f is_measurable_on E holds ex g being Functional_Sequence of X,ExtREAL st ( g is additive & ( for n being Nat holds ( g . n is_simple_func_in S & g . n is nonnegative & g . n is_measurable_on E ) ) & ( for x being Element of X st x in E holds ( g # x is summable & f . x = Sum (g # x) ) ) & ex I being ExtREAL_sequence st ( ( for n being Nat holds I . n = Integral (M,((g . n) | E)) ) & I is summable & Integral (M,(f | E)) = Sum I ) ) let M be sigma_Measure of S; ::_thesis: for E being Element of S for f being PartFunc of X,ExtREAL st E c= dom f & f is nonnegative & f is_measurable_on E holds ex g being Functional_Sequence of X,ExtREAL st ( g is additive & ( for n being Nat holds ( g . n is_simple_func_in S & g . n is nonnegative & g . n is_measurable_on E ) ) & ( for x being Element of X st x in E holds ( g # x is summable & f . x = Sum (g # x) ) ) & ex I being ExtREAL_sequence st ( ( for n being Nat holds I . n = Integral (M,((g . n) | E)) ) & I is summable & Integral (M,(f | E)) = Sum I ) ) let E be Element of S; ::_thesis: for f being PartFunc of X,ExtREAL st E c= dom f & f is nonnegative & f is_measurable_on E holds ex g being Functional_Sequence of X,ExtREAL st ( g is additive & ( for n being Nat holds ( g . n is_simple_func_in S & g . n is nonnegative & g . n is_measurable_on E ) ) & ( for x being Element of X st x in E holds ( g # x is summable & f . x = Sum (g # x) ) ) & ex I being ExtREAL_sequence st ( ( for n being Nat holds I . n = Integral (M,((g . n) | E)) ) & I is summable & Integral (M,(f | E)) = Sum I ) ) let f be PartFunc of X,ExtREAL; ::_thesis: ( E c= dom f & f is nonnegative & f is_measurable_on E implies ex g being Functional_Sequence of X,ExtREAL st ( g is additive & ( for n being Nat holds ( g . n is_simple_func_in S & g . n is nonnegative & g . n is_measurable_on E ) ) & ( for x being Element of X st x in E holds ( g # x is summable & f . x = Sum (g # x) ) ) & ex I being ExtREAL_sequence st ( ( for n being Nat holds I . n = Integral (M,((g . n) | E)) ) & I is summable & Integral (M,(f | E)) = Sum I ) ) ) assume that A1: E c= dom f and A2: f is nonnegative and A3: f is_measurable_on E ; ::_thesis: ex g being Functional_Sequence of X,ExtREAL st ( g is additive & ( for n being Nat holds ( g . n is_simple_func_in S & g . n is nonnegative & g . n is_measurable_on E ) ) & ( for x being Element of X st x in E holds ( g # x is summable & f . x = Sum (g # x) ) ) & ex I being ExtREAL_sequence st ( ( for n being Nat holds I . n = Integral (M,((g . n) | E)) ) & I is summable & Integral (M,(f | E)) = Sum I ) ) set F = f | E; A4: dom (f | E) = E by A1, RELAT_1:62; E = (dom f) /\ E by A1, XBOOLE_1:28; then f | E is_measurable_on E by A3, MESFUNC5:42; then consider h being Functional_Sequence of X,ExtREAL such that A5: for n being Nat holds ( h . n is_simple_func_in S & dom (h . n) = dom (f | E) ) and A6: for n being Nat holds h . n is nonnegative and A7: for n, m being Nat st n <= m holds for x being Element of X st x in dom (f | E) holds (h . n) . x <= (h . m) . x and A8: for x being Element of X st x in dom (f | E) holds ( h # x is convergent & lim (h # x) = (f | E) . x ) by A2, A4, MESFUNC5:15, MESFUNC5:64; defpred S1[ Element of NAT , set , set ] means $3 = (h . ($1 + 1)) - (h . $1); A9: for n being Element of NAT for x being set ex y being set st S1[n,x,y] ; consider g being Function such that A10: ( dom g = NAT & g . 0 = h . 0 & ( for n being Element of NAT holds S1[n,g . n,g . (n + 1)] ) ) from RECDEF_1:sch_1(A9); now__::_thesis:_for_f_being_set_st_f_in_rng_g_holds_ f_in_PFuncs_(X,ExtREAL) defpred S2[ Element of NAT ] means g . $1 is PartFunc of X,ExtREAL; let f be set ; ::_thesis: ( f in rng g implies f in PFuncs (X,ExtREAL) ) assume f in rng g ; ::_thesis: f in PFuncs (X,ExtREAL) then consider m being set such that A11: m in dom g and A12: f = g . m by FUNCT_1:def_3; reconsider m = m as Element of NAT by A10, A11; A13: for n being Element of NAT st S2[n] holds S2[n + 1] proof let n be Element of NAT ; ::_thesis: ( S2[n] implies S2[n + 1] ) assume S2[n] ; ::_thesis: S2[n + 1] g . (n + 1) = (h . (n + 1)) - (h . n) by A10; hence S2[n + 1] ; ::_thesis: verum end; A14: S2[ 0 ] by A10; for n being Element of NAT holds S2[n] from NAT_1:sch_1(A14, A13); then g . m is PartFunc of X,ExtREAL ; hence f in PFuncs (X,ExtREAL) by A12, PARTFUN1:45; ::_thesis: verum end; then rng g c= PFuncs (X,ExtREAL) by TARSKI:def_3; then reconsider g = g as Functional_Sequence of X,ExtREAL by A10, FUNCT_2:def_1, RELSET_1:4; take g ; ::_thesis: ( g is additive & ( for n being Nat holds ( g . n is_simple_func_in S & g . n is nonnegative & g . n is_measurable_on E ) ) & ( for x being Element of X st x in E holds ( g # x is summable & f . x = Sum (g # x) ) ) & ex I being ExtREAL_sequence st ( ( for n being Nat holds I . n = Integral (M,((g . n) | E)) ) & I is summable & Integral (M,(f | E)) = Sum I ) ) A15: for n being Nat holds ( g . n is_simple_func_in S & g . n is nonnegative & g . n is_measurable_on E & E c= dom (g . n) ) proof let n be Nat; ::_thesis: ( g . n is_simple_func_in S & g . n is nonnegative & g . n is_measurable_on E & E c= dom (g . n) ) percases ( n = 0 or n <> 0 ) ; supposeA16: n = 0 ; ::_thesis: ( g . n is_simple_func_in S & g . n is nonnegative & g . n is_measurable_on E & E c= dom (g . n) ) hence ( g . n is_simple_func_in S & g . n is nonnegative ) by A5, A6, A10; ::_thesis: ( g . n is_measurable_on E & E c= dom (g . n) ) hence g . n is_measurable_on E by MESFUNC2:34; ::_thesis: E c= dom (g . n) thus E c= dom (g . n) by A4, A5, A10, A16; ::_thesis: verum end; suppose n <> 0 ; ::_thesis: ( g . n is_simple_func_in S & g . n is nonnegative & g . n is_measurable_on E & E c= dom (g . n) ) then consider m being Nat such that A17: n = m + 1 by NAT_1:6; reconsider m = m as Element of NAT by ORDINAL1:def_12; A18: g . n = (h . n) - (h . m) by A10, A17; then A19: g . n = (h . n) + (- (h . m)) by MESFUNC2:8; A20: h . n is_simple_func_in S by A5; then A21: h . n is V119() by MESFUNC5:14; A22: dom (h . n) = dom (f | E) by A5; (- 1) (#) (h . m) is_simple_func_in S by A5, MESFUNC5:39; then A23: - (h . m) is_simple_func_in S by MESFUNC2:9; hence g . n is_simple_func_in S by A19, A20, MESFUNC5:38; ::_thesis: ( g . n is nonnegative & g . n is_measurable_on E & E c= dom (g . n) ) A24: h . m is_simple_func_in S by A5; then h . m is V120() by MESFUNC5:14; then A25: dom ((h . n) - (h . m)) = (dom (h . n)) /\ (dom (h . m)) by A21, MESFUNC5:17; A26: dom (h . m) = dom (f | E) by A5; then for x being set st x in dom ((h . n) - (h . m)) holds (h . m) . x <= (h . n) . x by A7, A17, A25, A22, NAT_1:11; hence g . n is nonnegative by A18, A20, A24, MESFUNC5:40; ::_thesis: ( g . n is_measurable_on E & E c= dom (g . n) ) thus g . n is_measurable_on E by A19, A20, A23, MESFUNC2:34, MESFUNC5:38; ::_thesis: E c= dom (g . n) thus E c= dom (g . n) by A4, A10, A17, A25, A22, A26; ::_thesis: verum end; end; end; hence A27: g is additive by Th35; ::_thesis: ( ( for n being Nat holds ( g . n is_simple_func_in S & g . n is nonnegative & g . n is_measurable_on E ) ) & ( for x being Element of X st x in E holds ( g # x is summable & f . x = Sum (g # x) ) ) & ex I being ExtREAL_sequence st ( ( for n being Nat holds I . n = Integral (M,((g . n) | E)) ) & I is summable & Integral (M,(f | E)) = Sum I ) ) thus for n being Nat holds ( g . n is_simple_func_in S & g . n is nonnegative & g . n is_measurable_on E ) by A15; ::_thesis: ( ( for x being Element of X st x in E holds ( g # x is summable & f . x = Sum (g # x) ) ) & ex I being ExtREAL_sequence st ( ( for n being Nat holds I . n = Integral (M,((g . n) | E)) ) & I is summable & Integral (M,(f | E)) = Sum I ) ) A28: now__::_thesis:_for_x_being_Element_of_X_st_x_in_E_holds_ (_g_#_x_is_summable_&_Sum_(g_#_x)_=_f_._x_) let x be Element of X; ::_thesis: ( x in E implies ( g # b1 is summable & Sum (g # b1) = f . b1 ) ) assume A29: x in E ; ::_thesis: ( g # b1 is summable & Sum (g # b1) = f . b1 ) then A30: h # x is convergent by A4, A8; A31: for m being Nat holds (Partial_Sums (g # x)) . m = (h # x) . m proof defpred S2[ Nat] means (Partial_Sums (g # x)) . $1 = (h # x) . $1; let m be Nat; ::_thesis: (Partial_Sums (g # x)) . m = (h # x) . m A32: for k being Nat st S2[k] holds S2[k + 1] proof set Pgx = Partial_Sums (g # x); let k be Nat; ::_thesis: ( S2[k] implies S2[k + 1] ) assume S2[k] ; ::_thesis: S2[k + 1] then A33: (Partial_Sums (g # x)) . k = (h . k) . x by MESFUNC5:def_13; A34: dom (h . (k + 1)) = dom (f | E) by A5; A35: h . (k + 1) is_simple_func_in S by A5; then A36: h . (k + 1) is V119() by MESFUNC5:14; then A37: -infty < (h . (k + 1)) . x by A4, A29, A34, MESFUNC5:10; h . (k + 1) is V120() by A35, MESFUNC5:14; then A38: (h . (k + 1)) . x < +infty by A4, A29, A34, MESFUNC5:11; A39: dom (h . k) = dom (f | E) by A5; A40: h . k is_simple_func_in S by A5; then A41: h . k is V120() by MESFUNC5:14; then A42: (h . k) . x < +infty by A4, A29, A39, MESFUNC5:11; h . k is V119() by A40, MESFUNC5:14; then A43: -infty < (h . k) . x by A4, A29, A39, MESFUNC5:10; reconsider k = k as Element of NAT by ORDINAL1:def_12; reconsider hk1x = (h . (k + 1)) . x as Real by A37, A38, XXREAL_0:14; A44: g . (k + 1) = (h . (k + 1)) - (h . k) by A10; reconsider hkx = (h . k) . x as Real by A43, A42, XXREAL_0:14; ((h . (k + 1)) . x) - ((h . k) . x) = hk1x - hkx by SUPINF_2:3; then A45: (((h . (k + 1)) . x) - ((h . k) . x)) + ((h . k) . x) = (hk1x - hkx) + hkx by SUPINF_2:1; (Partial_Sums (g # x)) . (k + 1) = ((Partial_Sums (g # x)) . k) + ((g # x) . (k + 1)) by Def1; then A46: (Partial_Sums (g # x)) . (k + 1) = ((h . k) . x) + ((g . (k + 1)) . x) by A33, MESFUNC5:def_13; dom ((h . (k + 1)) - (h . k)) = (dom (h . (k + 1))) /\ (dom (h . k)) by A36, A41, MESFUNC5:17; then (g . (k + 1)) . x = ((h . (k + 1)) . x) - ((h . k) . x) by A4, A29, A34, A39, A44, MESFUNC1:def_4; hence S2[k + 1] by A46, A45, MESFUNC5:def_13; ::_thesis: verum end; (Partial_Sums (g # x)) . 0 = (g # x) . 0 by Def1; then (Partial_Sums (g # x)) . 0 = (g . 0) . x by MESFUNC5:def_13; then A47: S2[ 0 ] by A10, MESFUNC5:def_13; for k being Nat holds S2[k] from NAT_1:sch_2(A47, A32); hence (Partial_Sums (g # x)) . m = (h # x) . m ; ::_thesis: verum end; A48: lim (h # x) = (f | E) . x by A4, A8, A29; percases ( h # x is convergent_to_finite_number or h # x is convergent_to_+infty or h # x is convergent_to_-infty ) by A30, MESFUNC5:def_11; supposeA49: h # x is convergent_to_finite_number ; ::_thesis: ( g # b1 is summable & Sum (g # b1) = f . b1 ) then A50: not h # x is convergent_to_-infty by MESFUNC5:51; not h # x is convergent_to_+infty by A49, MESFUNC5:50; then consider s being real number such that A51: lim (h # x) = s and A52: for p being real number st 0 < p holds ex N being Nat st for m being Nat st N <= m holds |.(((h # x) . m) - (lim (h # x))).| < p and h # x is convergent_to_finite_number by A30, A50, MESFUNC5:def_12; for p being real number st 0 < p holds ex N being Nat st for m being Nat st N <= m holds |.(((Partial_Sums (g # x)) . m) - (R_EAL s)).| < p proof let p be real number ; ::_thesis: ( 0 < p implies ex N being Nat st for m being Nat st N <= m holds |.(((Partial_Sums (g # x)) . m) - (R_EAL s)).| < p ) assume 0 < p ; ::_thesis: ex N being Nat st for m being Nat st N <= m holds |.(((Partial_Sums (g # x)) . m) - (R_EAL s)).| < p then consider N being Nat such that A53: for m being Nat st N <= m holds |.(((h # x) . m) - (lim (h # x))).| < p by A52; take N ; ::_thesis: for m being Nat st N <= m holds |.(((Partial_Sums (g # x)) . m) - (R_EAL s)).| < p let m be Nat; ::_thesis: ( N <= m implies |.(((Partial_Sums (g # x)) . m) - (R_EAL s)).| < p ) assume N <= m ; ::_thesis: |.(((Partial_Sums (g # x)) . m) - (R_EAL s)).| < p then |.(((h # x) . m) - (lim (h # x))).| < p by A53; hence |.(((Partial_Sums (g # x)) . m) - (R_EAL s)).| < p by A31, A51; ::_thesis: verum end; then A54: Partial_Sums (g # x) is convergent_to_finite_number by MESFUNC5:def_8; then A55: Partial_Sums (g # x) is convergent by MESFUNC5:def_11; hence g # x is summable by Def2; ::_thesis: Sum (g # x) = f . x for p being real number st 0 < p holds ex N being Nat st for m being Nat st N <= m holds |.(((Partial_Sums (g # x)) . m) - (lim (h # x))).| < p proof let p be real number ; ::_thesis: ( 0 < p implies ex N being Nat st for m being Nat st N <= m holds |.(((Partial_Sums (g # x)) . m) - (lim (h # x))).| < p ) assume 0 < p ; ::_thesis: ex N being Nat st for m being Nat st N <= m holds |.(((Partial_Sums (g # x)) . m) - (lim (h # x))).| < p then consider N being Nat such that A56: for m being Nat st N <= m holds |.(((h # x) . m) - (lim (h # x))).| < p by A52; take N ; ::_thesis: for m being Nat st N <= m holds |.(((Partial_Sums (g # x)) . m) - (lim (h # x))).| < p let m be Nat; ::_thesis: ( N <= m implies |.(((Partial_Sums (g # x)) . m) - (lim (h # x))).| < p ) assume N <= m ; ::_thesis: |.(((Partial_Sums (g # x)) . m) - (lim (h # x))).| < p then |.(((h # x) . m) - (lim (h # x))).| < p by A56; hence |.(((Partial_Sums (g # x)) . m) - (lim (h # x))).| < p by A31; ::_thesis: verum end; then lim (Partial_Sums (g # x)) = lim (h # x) by A51, A54, A55, MESFUNC5:def_12; hence Sum (g # x) = f . x by A29, A48, FUNCT_1:49; ::_thesis: verum end; supposeA57: h # x is convergent_to_+infty ; ::_thesis: ( g # b1 is summable & Sum (g # b1) = f . b1 ) for p being real number st 0 < p holds ex N being Nat st for m being Nat st N <= m holds p <= (Partial_Sums (g # x)) . m proof let p be real number ; ::_thesis: ( 0 < p implies ex N being Nat st for m being Nat st N <= m holds p <= (Partial_Sums (g # x)) . m ) assume 0 < p ; ::_thesis: ex N being Nat st for m being Nat st N <= m holds p <= (Partial_Sums (g # x)) . m then consider N being Nat such that A58: for m being Nat st N <= m holds p <= (h # x) . m by A57, MESFUNC5:def_9; take N ; ::_thesis: for m being Nat st N <= m holds p <= (Partial_Sums (g # x)) . m let m be Nat; ::_thesis: ( N <= m implies p <= (Partial_Sums (g # x)) . m ) assume N <= m ; ::_thesis: p <= (Partial_Sums (g # x)) . m then p <= (h # x) . m by A58; hence p <= (Partial_Sums (g # x)) . m by A31; ::_thesis: verum end; then A59: Partial_Sums (g # x) is convergent_to_+infty by MESFUNC5:def_9; then A60: Partial_Sums (g # x) is convergent by MESFUNC5:def_11; then A61: lim (Partial_Sums (g # x)) = +infty by A59, MESFUNC5:def_12; thus g # x is summable by A60, Def2; ::_thesis: Sum (g # x) = f . x lim (h # x) = +infty by A30, A57, MESFUNC5:def_12; hence Sum (g # x) = f . x by A29, A48, A61, FUNCT_1:49; ::_thesis: verum end; supposeA62: h # x is convergent_to_-infty ; ::_thesis: ( g # b1 is summable & Sum (g # b1) = f . b1 ) for p being real number st p < 0 holds ex N being Nat st for m being Nat st N <= m holds (Partial_Sums (g # x)) . m <= p proof let p be real number ; ::_thesis: ( p < 0 implies ex N being Nat st for m being Nat st N <= m holds (Partial_Sums (g # x)) . m <= p ) assume p < 0 ; ::_thesis: ex N being Nat st for m being Nat st N <= m holds (Partial_Sums (g # x)) . m <= p then consider N being Nat such that A63: for m being Nat st N <= m holds (h # x) . m <= p by A62, MESFUNC5:def_10; take N ; ::_thesis: for m being Nat st N <= m holds (Partial_Sums (g # x)) . m <= p let m be Nat; ::_thesis: ( N <= m implies (Partial_Sums (g # x)) . m <= p ) assume N <= m ; ::_thesis: (Partial_Sums (g # x)) . m <= p then (h # x) . m <= p by A63; hence (Partial_Sums (g # x)) . m <= p by A31; ::_thesis: verum end; then A64: Partial_Sums (g # x) is convergent_to_-infty by MESFUNC5:def_10; then A65: Partial_Sums (g # x) is convergent by MESFUNC5:def_11; then A66: lim (Partial_Sums (g # x)) = -infty by A64, MESFUNC5:def_12; thus g # x is summable by A65, Def2; ::_thesis: Sum (g # x) = f . x lim (h # x) = -infty by A30, A62, MESFUNC5:def_12; hence Sum (g # x) = f . x by A29, A48, A66, FUNCT_1:49; ::_thesis: verum end; end; end; hence for x being Element of X st x in E holds ( g # x is summable & f . x = Sum (g # x) ) ; ::_thesis: ex I being ExtREAL_sequence st ( ( for n being Nat holds I . n = Integral (M,((g . n) | E)) ) & I is summable & Integral (M,(f | E)) = Sum I ) percases ( E = {} or E <> {} ) ; suppose E = {} ; ::_thesis: ex I being ExtREAL_sequence st ( ( for n being Nat holds I . n = Integral (M,((g . n) | E)) ) & I is summable & Integral (M,(f | E)) = Sum I ) hence ex I being ExtREAL_sequence st ( ( for n being Nat holds I . n = Integral (M,((g . n) | E)) ) & I is summable & Integral (M,(f | E)) = Sum I ) by A1, A3, A15, Lm2; ::_thesis: verum end; supposeA67: E <> {} ; ::_thesis: ex I being ExtREAL_sequence st ( ( for n being Nat holds I . n = Integral (M,((g . n) | E)) ) & I is summable & Integral (M,(f | E)) = Sum I ) for m being Element of NAT holds ( g . m is_simple_func_in S & g . m is nonnegative & g . m is_measurable_on E & E c= dom (g . m) ) by A15; then E common_on_dom g by A67, SEQFUNC:def_9; hence ex I being ExtREAL_sequence st ( ( for n being Nat holds I . n = Integral (M,((g . n) | E)) ) & I is summable & Integral (M,(f | E)) = Sum I ) by A1, A2, A3, A15, A27, A28, Lm3; ::_thesis: verum end; end; end; registration let X be non empty set ; cluster non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total with_the_same_dom additive for Element of bool [:NAT,(PFuncs (X,ExtREAL)):]; existence ex b1 being Functional_Sequence of X,ExtREAL st ( b1 is additive & b1 is with_the_same_dom ) proof deffunc H1( Nat) -> Element of bool [:X,ExtREAL:] = <:{},X,ExtREAL:>; consider F being Functional_Sequence of X,ExtREAL such that A1: for n being Nat holds F . n = H1(n) from SEQFUNC:sch_1(); now__::_thesis:_for_n,_m_being_Nat_holds_dom_(F_._n)_=_dom_(F_._m) let n, m be Nat; ::_thesis: dom (F . n) = dom (F . m) F . n = <:{},X,ExtREAL:> by A1; hence dom (F . n) = dom (F . m) by A1; ::_thesis: verum end; then reconsider F = F as with_the_same_dom Functional_Sequence of X,ExtREAL by MESFUNC8:def_2; take F ; ::_thesis: ( F is additive & F is with_the_same_dom ) now__::_thesis:_for_n,_m_being_Nat_st_n_<>_m_holds_ for_x_being_set_holds_ (_not_x_in_(dom_(F_._n))_/\_(dom_(F_._m))_or_(F_._n)_._x_<>_+infty_or_(F_._m)_._x_<>_-infty_) let n, m be Nat; ::_thesis: ( n <> m implies for x being set holds ( not x in (dom (F . n)) /\ (dom (F . m)) or (F . n) . x <> +infty or (F . m) . x <> -infty ) ) assume n <> m ; ::_thesis: for x being set holds ( not x in (dom (F . n)) /\ (dom (F . m)) or (F . n) . x <> +infty or (F . m) . x <> -infty ) let x be set ; ::_thesis: ( not x in (dom (F . n)) /\ (dom (F . m)) or (F . n) . x <> +infty or (F . m) . x <> -infty ) assume A2: x in (dom (F . n)) /\ (dom (F . m)) ; ::_thesis: ( (F . n) . x <> +infty or (F . m) . x <> -infty ) F . n = <:{},X,ExtREAL:> by A1; then x in (dom {}) /\ (dom (F . m)) by A2, PARTFUN1:34; hence ( (F . n) . x <> +infty or (F . m) . x <> -infty ) ; ::_thesis: verum end; hence ( F is additive & F is with_the_same_dom ) by Def5; ::_thesis: verum end; end; definition let C, D, X be non empty set ; let F be Function of [:C,D:],(PFuncs (X,ExtREAL)); let c be Element of C; let d be Element of D; :: original: . redefine funcF . (c,d) -> PartFunc of X,ExtREAL; correctness coherence F . (c,d) is PartFunc of X,ExtREAL; proof thus F . (c,d) is PartFunc of X,ExtREAL by PARTFUN1:47; ::_thesis: verum end; end; definition let C, D, X be non empty set ; let F be Function of [:C,D:],X; let c be Element of C; func ProjMap1 (F,c) -> Function of D,X means :: MESFUNC9:def 6 for d being Element of D holds it . d = F . (c,d); existence ex b1 being Function of D,X st for d being Element of D holds b1 . d = F . (c,d) proof deffunc H1( Element of D) -> Element of X = F . (c,$1); consider IT being Function such that A1: ( dom IT = D & ( for d being Element of D holds IT . d = H1(d) ) ) from FUNCT_1:sch_4(); now__::_thesis:_for_d_being_set_st_d_in_D_holds_ IT_._d_in_X let d be set ; ::_thesis: ( d in D implies IT . d in X ) assume A2: d in D ; ::_thesis: IT . d in X then A3: [c,d] in [:C,D:] by ZFMISC_1:87; IT . d = F . (c,d) by A1, A2; hence IT . d in X by A3, FUNCT_2:5; ::_thesis: verum end; then reconsider IT = IT as Function of D,X by A1, FUNCT_2:3; take IT ; ::_thesis: for d being Element of D holds IT . d = F . (c,d) let d be Element of D; ::_thesis: IT . d = F . (c,d) thus IT . d = F . (c,d) by A1; ::_thesis: verum end; uniqueness for b1, b2 being Function of D,X st ( for d being Element of D holds b1 . d = F . (c,d) ) & ( for d being Element of D holds b2 . d = F . (c,d) ) holds b1 = b2 proof let P1, P2 be Function of D,X; ::_thesis: ( ( for d being Element of D holds P1 . d = F . (c,d) ) & ( for d being Element of D holds P2 . d = F . (c,d) ) implies P1 = P2 ) assume that A4: for d being Element of D holds P1 . d = F . (c,d) and A5: for d being Element of D holds P2 . d = F . (c,d) ; ::_thesis: P1 = P2 now__::_thesis:_for_d_being_set_st_d_in_D_holds_ P1_._d_=_P2_._d let d be set ; ::_thesis: ( d in D implies P1 . d = P2 . d ) assume d in D ; ::_thesis: P1 . d = P2 . d then reconsider d1 = d as Element of D ; P1 . d1 = F . (c,d1) by A4; hence P1 . d = P2 . d by A5; ::_thesis: verum end; hence P1 = P2 by FUNCT_2:12; ::_thesis: verum end; end; :: deftheorem defines ProjMap1 MESFUNC9:def_6_:_ for C, D, X being non empty set for F being Function of [:C,D:],X for c being Element of C for b6 being Function of D,X holds ( b6 = ProjMap1 (F,c) iff for d being Element of D holds b6 . d = F . (c,d) ); definition let C, D, X be non empty set ; let F be Function of [:C,D:],X; let d be Element of D; func ProjMap2 (F,d) -> Function of C,X means :: MESFUNC9:def 7 for c being Element of C holds it . c = F . (c,d); existence ex b1 being Function of C,X st for c being Element of C holds b1 . c = F . (c,d) proof deffunc H1( Element of C) -> Element of X = F . ($1,d); consider IT being Function such that A1: ( dom IT = C & ( for c being Element of C holds IT . c = H1(c) ) ) from FUNCT_1:sch_4(); now__::_thesis:_for_c_being_set_st_c_in_C_holds_ IT_._c_in_X let c be set ; ::_thesis: ( c in C implies IT . c in X ) assume A2: c in C ; ::_thesis: IT . c in X then A3: [c,d] in [:C,D:] by ZFMISC_1:87; IT . c = F . (c,d) by A1, A2 .= F . [c,d] ; hence IT . c in X by A3, FUNCT_2:5; ::_thesis: verum end; then reconsider IT = IT as Function of C,X by A1, FUNCT_2:3; take IT ; ::_thesis: for c being Element of C holds IT . c = F . (c,d) let c be Element of C; ::_thesis: IT . c = F . (c,d) thus IT . c = F . (c,d) by A1; ::_thesis: verum end; uniqueness for b1, b2 being Function of C,X st ( for c being Element of C holds b1 . c = F . (c,d) ) & ( for c being Element of C holds b2 . c = F . (c,d) ) holds b1 = b2 proof let P1, P2 be Function of C,X; ::_thesis: ( ( for c being Element of C holds P1 . c = F . (c,d) ) & ( for c being Element of C holds P2 . c = F . (c,d) ) implies P1 = P2 ) assume that A4: for c being Element of C holds P1 . c = F . (c,d) and A5: for c being Element of C holds P2 . c = F . (c,d) ; ::_thesis: P1 = P2 now__::_thesis:_for_c_being_set_st_c_in_C_holds_ P1_._c_=_P2_._c let c be set ; ::_thesis: ( c in C implies P1 . c = P2 . c ) assume c in C ; ::_thesis: P1 . c = P2 . c then reconsider c1 = c as Element of C ; P1 . c1 = F . (c1,d) by A4; hence P1 . c = P2 . c by A5; ::_thesis: verum end; hence P1 = P2 by FUNCT_2:12; ::_thesis: verum end; end; :: deftheorem defines ProjMap2 MESFUNC9:def_7_:_ for C, D, X being non empty set for F being Function of [:C,D:],X for d being Element of D for b6 being Function of C,X holds ( b6 = ProjMap2 (F,d) iff for c being Element of C holds b6 . c = F . (c,d) ); definition let X, Y be set ; let F be Function of [:NAT,NAT:],(PFuncs (X,Y)); let n be Nat; func ProjMap1 (F,n) -> Functional_Sequence of X,Y means :Def8: :: MESFUNC9:def 8 for m being Nat holds it . m = F . (n,m); existence ex b1 being Functional_Sequence of X,Y st for m being Nat holds b1 . m = F . (n,m) proof deffunc H1( Element of NAT ) -> set = F . (n,$1); consider IT being Function such that A1: ( dom IT = NAT & ( for m being Element of NAT holds IT . m = H1(m) ) ) from FUNCT_1:sch_4(); now__::_thesis:_for_y_being_set_st_y_in_rng_IT_holds_ y_in_PFuncs_(X,Y) reconsider n1 = n as Element of NAT by ORDINAL1:def_12; let y be set ; ::_thesis: ( y in rng IT implies y in PFuncs (X,Y) ) assume y in rng IT ; ::_thesis: y in PFuncs (X,Y) then consider k being set such that A2: k in dom IT and A3: y = IT . k by FUNCT_1:def_3; reconsider k = k as Element of NAT by A1, A2; y = F . (n1,k) by A1, A3; hence y in PFuncs (X,Y) ; ::_thesis: verum end; then rng IT c= PFuncs (X,Y) by TARSKI:def_3; then reconsider IT = IT as Functional_Sequence of X,Y by A1, FUNCT_2:def_1, RELSET_1:4; take IT ; ::_thesis: for m being Nat holds IT . m = F . (n,m) thus for m being Nat holds IT . m = F . (n,m) ::_thesis: verum proof let m be Nat; ::_thesis: IT . m = F . (n,m) reconsider n1 = n, m1 = m as Element of NAT by ORDINAL1:def_12; IT . m = F . (n1,m1) by A1; hence IT . m = F . (n,m) ; ::_thesis: verum end; end; uniqueness for b1, b2 being Functional_Sequence of X,Y st ( for m being Nat holds b1 . m = F . (n,m) ) & ( for m being Nat holds b2 . m = F . (n,m) ) holds b1 = b2 proof let G1, G2 be Functional_Sequence of X,Y; ::_thesis: ( ( for m being Nat holds G1 . m = F . (n,m) ) & ( for m being Nat holds G2 . m = F . (n,m) ) implies G1 = G2 ) assume that A4: for m being Nat holds G1 . m = F . (n,m) and A5: for m being Nat holds G2 . m = F . (n,m) ; ::_thesis: G1 = G2 for m being Element of NAT holds G1 . m = G2 . m proof let m be Element of NAT ; ::_thesis: G1 . m = G2 . m reconsider m1 = m as Nat ; G1 . m = F . (n,m1) by A4; hence G1 . m = G2 . m by A5; ::_thesis: verum end; hence G1 = G2 by FUNCT_2:63; ::_thesis: verum end; func ProjMap2 (F,n) -> Functional_Sequence of X,Y means :Def9: :: MESFUNC9:def 9 for m being Nat holds it . m = F . (m,n); existence ex b1 being Functional_Sequence of X,Y st for m being Nat holds b1 . m = F . (m,n) proof deffunc H1( Element of NAT ) -> set = F . ($1,n); consider IT being Function such that A6: ( dom IT = NAT & ( for m being Element of NAT holds IT . m = H1(m) ) ) from FUNCT_1:sch_4(); now__::_thesis:_for_y_being_set_st_y_in_rng_IT_holds_ y_in_PFuncs_(X,Y) reconsider n1 = n as Element of NAT by ORDINAL1:def_12; let y be set ; ::_thesis: ( y in rng IT implies y in PFuncs (X,Y) ) assume y in rng IT ; ::_thesis: y in PFuncs (X,Y) then consider k being set such that A7: k in dom IT and A8: y = IT . k by FUNCT_1:def_3; reconsider k = k as Element of NAT by A6, A7; y = F . (k,n1) by A6, A8; hence y in PFuncs (X,Y) ; ::_thesis: verum end; then rng IT c= PFuncs (X,Y) by TARSKI:def_3; then reconsider IT = IT as Functional_Sequence of X,Y by A6, FUNCT_2:def_1, RELSET_1:4; take IT ; ::_thesis: for m being Nat holds IT . m = F . (m,n) thus for m being Nat holds IT . m = F . (m,n) ::_thesis: verum proof let m be Nat; ::_thesis: IT . m = F . (m,n) reconsider n1 = n, m1 = m as Element of NAT by ORDINAL1:def_12; IT . m = F . (m1,n1) by A6; hence IT . m = F . (m,n) ; ::_thesis: verum end; end; uniqueness for b1, b2 being Functional_Sequence of X,Y st ( for m being Nat holds b1 . m = F . (m,n) ) & ( for m being Nat holds b2 . m = F . (m,n) ) holds b1 = b2 proof let G1, G2 be Functional_Sequence of X,Y; ::_thesis: ( ( for m being Nat holds G1 . m = F . (m,n) ) & ( for m being Nat holds G2 . m = F . (m,n) ) implies G1 = G2 ) assume that A9: for m being Nat holds G1 . m = F . (m,n) and A10: for m being Nat holds G2 . m = F . (m,n) ; ::_thesis: G1 = G2 for m being Element of NAT holds G1 . m = G2 . m proof let m be Element of NAT ; ::_thesis: G1 . m = G2 . m reconsider m1 = m as Nat ; G1 . m = F . (m1,n) by A9; hence G1 . m = G2 . m by A10; ::_thesis: verum end; hence G1 = G2 by FUNCT_2:63; ::_thesis: verum end; end; :: deftheorem Def8 defines ProjMap1 MESFUNC9:def_8_:_ for X, Y being set for F being Function of [:NAT,NAT:],(PFuncs (X,Y)) for n being Nat for b5 being Functional_Sequence of X,Y holds ( b5 = ProjMap1 (F,n) iff for m being Nat holds b5 . m = F . (n,m) ); :: deftheorem Def9 defines ProjMap2 MESFUNC9:def_9_:_ for X, Y being set for F being Function of [:NAT,NAT:],(PFuncs (X,Y)) for n being Nat for b5 being Functional_Sequence of X,Y holds ( b5 = ProjMap2 (F,n) iff for m being Nat holds b5 . m = F . (m,n) ); definition let X be non empty set ; let F be Function of NAT,(Funcs (NAT,(PFuncs (X,ExtREAL)))); let n be Nat; :: original: . redefine funcF . n -> Functional_Sequence of X,ExtREAL; correctness coherence F . n is Functional_Sequence of X,ExtREAL; proof ex f being Function st ( F . n = f & dom f = NAT & rng f c= PFuncs (X,ExtREAL) ) by FUNCT_2:def_2; hence F . n is Functional_Sequence of X,ExtREAL by FUNCT_2:def_1, RELSET_1:4; ::_thesis: verum end; end; theorem Th49: :: MESFUNC9:49 for X being non empty set for S being SigmaField of X for E being Element of S for F being Functional_Sequence of X,ExtREAL st E = dom (F . 0) & F is with_the_same_dom & ( for n being Nat holds ( F . n is nonnegative & F . n is_measurable_on E ) ) holds ex FF being Function of NAT,(Funcs (NAT,(PFuncs (X,ExtREAL)))) st for n being Nat holds ( ( for m being Nat holds ( (FF . n) . m is_simple_func_in S & dom ((FF . n) . m) = dom (F . n) ) ) & ( for m being Nat holds (FF . n) . m is nonnegative ) & ( for j, k being Nat st j <= k holds for x being Element of X st x in dom (F . n) holds ((FF . n) . j) . x <= ((FF . n) . k) . x ) & ( for x being Element of X st x in dom (F . n) holds ( (FF . n) # x is convergent & lim ((FF . n) # x) = (F . n) . x ) ) ) proof let X be non empty set ; ::_thesis: for S being SigmaField of X for E being Element of S for F being Functional_Sequence of X,ExtREAL st E = dom (F . 0) & F is with_the_same_dom & ( for n being Nat holds ( F . n is nonnegative & F . n is_measurable_on E ) ) holds ex FF being Function of NAT,(Funcs (NAT,(PFuncs (X,ExtREAL)))) st for n being Nat holds ( ( for m being Nat holds ( (FF . n) . m is_simple_func_in S & dom ((FF . n) . m) = dom (F . n) ) ) & ( for m being Nat holds (FF . n) . m is nonnegative ) & ( for j, k being Nat st j <= k holds for x being Element of X st x in dom (F . n) holds ((FF . n) . j) . x <= ((FF . n) . k) . x ) & ( for x being Element of X st x in dom (F . n) holds ( (FF . n) # x is convergent & lim ((FF . n) # x) = (F . n) . x ) ) ) let S be SigmaField of X; ::_thesis: for E being Element of S for F being Functional_Sequence of X,ExtREAL st E = dom (F . 0) & F is with_the_same_dom & ( for n being Nat holds ( F . n is nonnegative & F . n is_measurable_on E ) ) holds ex FF being Function of NAT,(Funcs (NAT,(PFuncs (X,ExtREAL)))) st for n being Nat holds ( ( for m being Nat holds ( (FF . n) . m is_simple_func_in S & dom ((FF . n) . m) = dom (F . n) ) ) & ( for m being Nat holds (FF . n) . m is nonnegative ) & ( for j, k being Nat st j <= k holds for x being Element of X st x in dom (F . n) holds ((FF . n) . j) . x <= ((FF . n) . k) . x ) & ( for x being Element of X st x in dom (F . n) holds ( (FF . n) # x is convergent & lim ((FF . n) # x) = (F . n) . x ) ) ) let E be Element of S; ::_thesis: for F being Functional_Sequence of X,ExtREAL st E = dom (F . 0) & F is with_the_same_dom & ( for n being Nat holds ( F . n is nonnegative & F . n is_measurable_on E ) ) holds ex FF being Function of NAT,(Funcs (NAT,(PFuncs (X,ExtREAL)))) st for n being Nat holds ( ( for m being Nat holds ( (FF . n) . m is_simple_func_in S & dom ((FF . n) . m) = dom (F . n) ) ) & ( for m being Nat holds (FF . n) . m is nonnegative ) & ( for j, k being Nat st j <= k holds for x being Element of X st x in dom (F . n) holds ((FF . n) . j) . x <= ((FF . n) . k) . x ) & ( for x being Element of X st x in dom (F . n) holds ( (FF . n) # x is convergent & lim ((FF . n) # x) = (F . n) . x ) ) ) let F be Functional_Sequence of X,ExtREAL; ::_thesis: ( E = dom (F . 0) & F is with_the_same_dom & ( for n being Nat holds ( F . n is nonnegative & F . n is_measurable_on E ) ) implies ex FF being Function of NAT,(Funcs (NAT,(PFuncs (X,ExtREAL)))) st for n being Nat holds ( ( for m being Nat holds ( (FF . n) . m is_simple_func_in S & dom ((FF . n) . m) = dom (F . n) ) ) & ( for m being Nat holds (FF . n) . m is nonnegative ) & ( for j, k being Nat st j <= k holds for x being Element of X st x in dom (F . n) holds ((FF . n) . j) . x <= ((FF . n) . k) . x ) & ( for x being Element of X st x in dom (F . n) holds ( (FF . n) # x is convergent & lim ((FF . n) # x) = (F . n) . x ) ) ) ) assume that A1: E = dom (F . 0) and A2: F is with_the_same_dom and A3: for n being Nat holds ( F . n is nonnegative & F . n is_measurable_on E ) ; ::_thesis: ex FF being Function of NAT,(Funcs (NAT,(PFuncs (X,ExtREAL)))) st for n being Nat holds ( ( for m being Nat holds ( (FF . n) . m is_simple_func_in S & dom ((FF . n) . m) = dom (F . n) ) ) & ( for m being Nat holds (FF . n) . m is nonnegative ) & ( for j, k being Nat st j <= k holds for x being Element of X st x in dom (F . n) holds ((FF . n) . j) . x <= ((FF . n) . k) . x ) & ( for x being Element of X st x in dom (F . n) holds ( (FF . n) # x is convergent & lim ((FF . n) # x) = (F . n) . x ) ) ) defpred S1[ Element of NAT , set ] means for G being Functional_Sequence of X,ExtREAL st $2 = G holds ( ( for m being Nat holds ( G . m is_simple_func_in S & dom (G . m) = dom (F . $1) ) ) & ( for m being Nat holds G . m is nonnegative ) & ( for j, k being Nat st j <= k holds for x being Element of X st x in dom (F . $1) holds (G . j) . x <= (G . k) . x ) & ( for x being Element of X st x in dom (F . $1) holds ( G # x is convergent & lim (G # x) = (F . $1) . x ) ) ); A4: for n being Element of NAT ex G being Functional_Sequence of X,ExtREAL st ( ( for m being Nat holds ( G . m is_simple_func_in S & dom (G . m) = dom (F . n) ) ) & ( for m being Nat holds G . m is nonnegative ) & ( for j, k being Nat st j <= k holds for x being Element of X st x in dom (F . n) holds (G . j) . x <= (G . k) . x ) & ( for x being Element of X st x in dom (F . n) holds ( G # x is convergent & lim (G # x) = (F . n) . x ) ) ) proof let n be Element of NAT ; ::_thesis: ex G being Functional_Sequence of X,ExtREAL st ( ( for m being Nat holds ( G . m is_simple_func_in S & dom (G . m) = dom (F . n) ) ) & ( for m being Nat holds G . m is nonnegative ) & ( for j, k being Nat st j <= k holds for x being Element of X st x in dom (F . n) holds (G . j) . x <= (G . k) . x ) & ( for x being Element of X st x in dom (F . n) holds ( G # x is convergent & lim (G # x) = (F . n) . x ) ) ) A5: F . n is_measurable_on E by A3; A6: F . n is nonnegative by A3; E = dom (F . n) by A1, A2, MESFUNC8:def_2; hence ex G being Functional_Sequence of X,ExtREAL st ( ( for m being Nat holds ( G . m is_simple_func_in S & dom (G . m) = dom (F . n) ) ) & ( for m being Nat holds G . m is nonnegative ) & ( for j, k being Nat st j <= k holds for x being Element of X st x in dom (F . n) holds (G . j) . x <= (G . k) . x ) & ( for x being Element of X st x in dom (F . n) holds ( G # x is convergent & lim (G # x) = (F . n) . x ) ) ) by A5, A6, MESFUNC5:64; ::_thesis: verum end; A7: for n being Element of NAT ex G being Element of Funcs (NAT,(PFuncs (X,ExtREAL))) st S1[n,G] proof let n be Element of NAT ; ::_thesis: ex G being Element of Funcs (NAT,(PFuncs (X,ExtREAL))) st S1[n,G] consider G being Functional_Sequence of X,ExtREAL such that A8: for m being Nat holds ( G . m is_simple_func_in S & dom (G . m) = dom (F . n) ) and A9: for m being Nat holds G . m is nonnegative and A10: for j, k being Nat st j <= k holds for x being Element of X st x in dom (F . n) holds (G . j) . x <= (G . k) . x and A11: for x being Element of X st x in dom (F . n) holds ( G # x is convergent & lim (G # x) = (F . n) . x ) by A4; reconsider G = G as Element of Funcs (NAT,(PFuncs (X,ExtREAL))) by FUNCT_2:8; take G ; ::_thesis: S1[n,G] thus S1[n,G] by A8, A9, A10, A11; ::_thesis: verum end; consider FF being Function of NAT,(Funcs (NAT,(PFuncs (X,ExtREAL)))) such that A12: for n being Element of NAT holds S1[n,FF . n] from FUNCT_2:sch_3(A7); take FF ; ::_thesis: for n being Nat holds ( ( for m being Nat holds ( (FF . n) . m is_simple_func_in S & dom ((FF . n) . m) = dom (F . n) ) ) & ( for m being Nat holds (FF . n) . m is nonnegative ) & ( for j, k being Nat st j <= k holds for x being Element of X st x in dom (F . n) holds ((FF . n) . j) . x <= ((FF . n) . k) . x ) & ( for x being Element of X st x in dom (F . n) holds ( (FF . n) # x is convergent & lim ((FF . n) # x) = (F . n) . x ) ) ) thus for n being Nat holds ( ( for m being Nat holds ( (FF . n) . m is_simple_func_in S & dom ((FF . n) . m) = dom (F . n) ) ) & ( for m being Nat holds (FF . n) . m is nonnegative ) & ( for j, k being Nat st j <= k holds for x being Element of X st x in dom (F . n) holds ((FF . n) . j) . x <= ((FF . n) . k) . x ) & ( for x being Element of X st x in dom (F . n) holds ( (FF . n) # x is convergent & lim ((FF . n) # x) = (F . n) . x ) ) ) ::_thesis: verum proof let n be Nat; ::_thesis: ( ( for m being Nat holds ( (FF . n) . m is_simple_func_in S & dom ((FF . n) . m) = dom (F . n) ) ) & ( for m being Nat holds (FF . n) . m is nonnegative ) & ( for j, k being Nat st j <= k holds for x being Element of X st x in dom (F . n) holds ((FF . n) . j) . x <= ((FF . n) . k) . x ) & ( for x being Element of X st x in dom (F . n) holds ( (FF . n) # x is convergent & lim ((FF . n) # x) = (F . n) . x ) ) ) reconsider n1 = n as Element of NAT by ORDINAL1:def_12; for G being Functional_Sequence of X,ExtREAL st FF . n1 = G holds ( ( for m being Nat holds ( G . m is_simple_func_in S & dom (G . m) = dom (F . n1) ) ) & ( for m being Nat holds G . m is nonnegative ) & ( for j, k being Nat st j <= k holds for x being Element of X st x in dom (F . n1) holds (G . j) . x <= (G . k) . x ) & ( for x being Element of X st x in dom (F . n1) holds ( G # x is convergent & lim (G # x) = (F . n1) . x ) ) ) by A12; hence ( ( for m being Nat holds ( (FF . n) . m is_simple_func_in S & dom ((FF . n) . m) = dom (F . n) ) ) & ( for m being Nat holds (FF . n) . m is nonnegative ) & ( for j, k being Nat st j <= k holds for x being Element of X st x in dom (F . n) holds ((FF . n) . j) . x <= ((FF . n) . k) . x ) & ( for x being Element of X st x in dom (F . n) holds ( (FF . n) # x is convergent & lim ((FF . n) # x) = (F . n) . x ) ) ) ; ::_thesis: verum end; end; theorem Th50: :: MESFUNC9:50 for X being non empty set for S being SigmaField of X for M being sigma_Measure of S for E being Element of S for F being Functional_Sequence of X,ExtREAL st E = dom (F . 0) & F is additive & F is with_the_same_dom & ( for n being Nat holds ( F . n is_measurable_on E & F . n is nonnegative ) ) holds ex I being ExtREAL_sequence st for n being Nat holds ( I . n = Integral (M,(F . n)) & Integral (M,((Partial_Sums F) . n)) = (Partial_Sums I) . n ) proof let X be non empty set ; ::_thesis: for S being SigmaField of X for M being sigma_Measure of S for E being Element of S for F being Functional_Sequence of X,ExtREAL st E = dom (F . 0) & F is additive & F is with_the_same_dom & ( for n being Nat holds ( F . n is_measurable_on E & F . n is nonnegative ) ) holds ex I being ExtREAL_sequence st for n being Nat holds ( I . n = Integral (M,(F . n)) & Integral (M,((Partial_Sums F) . n)) = (Partial_Sums I) . n ) let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S for E being Element of S for F being Functional_Sequence of X,ExtREAL st E = dom (F . 0) & F is additive & F is with_the_same_dom & ( for n being Nat holds ( F . n is_measurable_on E & F . n is nonnegative ) ) holds ex I being ExtREAL_sequence st for n being Nat holds ( I . n = Integral (M,(F . n)) & Integral (M,((Partial_Sums F) . n)) = (Partial_Sums I) . n ) let M be sigma_Measure of S; ::_thesis: for E being Element of S for F being Functional_Sequence of X,ExtREAL st E = dom (F . 0) & F is additive & F is with_the_same_dom & ( for n being Nat holds ( F . n is_measurable_on E & F . n is nonnegative ) ) holds ex I being ExtREAL_sequence st for n being Nat holds ( I . n = Integral (M,(F . n)) & Integral (M,((Partial_Sums F) . n)) = (Partial_Sums I) . n ) let E be Element of S; ::_thesis: for F being Functional_Sequence of X,ExtREAL st E = dom (F . 0) & F is additive & F is with_the_same_dom & ( for n being Nat holds ( F . n is_measurable_on E & F . n is nonnegative ) ) holds ex I being ExtREAL_sequence st for n being Nat holds ( I . n = Integral (M,(F . n)) & Integral (M,((Partial_Sums F) . n)) = (Partial_Sums I) . n ) let F be Functional_Sequence of X,ExtREAL; ::_thesis: ( E = dom (F . 0) & F is additive & F is with_the_same_dom & ( for n being Nat holds ( F . n is_measurable_on E & F . n is nonnegative ) ) implies ex I being ExtREAL_sequence st for n being Nat holds ( I . n = Integral (M,(F . n)) & Integral (M,((Partial_Sums F) . n)) = (Partial_Sums I) . n ) ) assume that A1: E = dom (F . 0) and A2: F is additive and A3: F is with_the_same_dom and A4: for n being Nat holds ( F . n is_measurable_on E & F . n is nonnegative ) ; ::_thesis: ex I being ExtREAL_sequence st for n being Nat holds ( I . n = Integral (M,(F . n)) & Integral (M,((Partial_Sums F) . n)) = (Partial_Sums I) . n ) set PF = Partial_Sums F; deffunc H1( Element of NAT ) -> Element of ExtREAL = Integral (M,(F . $1)); consider I being Function of NAT,ExtREAL such that A5: for n being Element of NAT holds I . n = H1(n) from FUNCT_2:sch_4(); reconsider I = I as ExtREAL_sequence ; take I ; ::_thesis: for n being Nat holds ( I . n = Integral (M,(F . n)) & Integral (M,((Partial_Sums F) . n)) = (Partial_Sums I) . n ) A6: for n being Nat holds F . n is V119() by A4, MESFUNC5:12; thus for n being Nat holds ( I . n = Integral (M,(F . n)) & Integral (M,((Partial_Sums F) . n)) = (Partial_Sums I) . n ) ::_thesis: verum proof set PI = Partial_Sums I; defpred S1[ Nat] means Integral (M,((Partial_Sums F) . $1)) = (Partial_Sums I) . $1; let n be Nat; ::_thesis: ( I . n = Integral (M,(F . n)) & Integral (M,((Partial_Sums F) . n)) = (Partial_Sums I) . n ) reconsider n9 = n as Element of NAT by ORDINAL1:def_12; I . n = Integral (M,(F . n9)) by A5; hence I . n = Integral (M,(F . n)) ; ::_thesis: Integral (M,((Partial_Sums F) . n)) = (Partial_Sums I) . n A7: for k being Nat st S1[k] holds S1[k + 1] proof let k be Nat; ::_thesis: ( S1[k] implies S1[k + 1] ) assume A8: S1[k] ; ::_thesis: S1[k + 1] A9: F . (k + 1) is_measurable_on E by A4; A10: dom (F . (k + 1)) = E by A1, A3, MESFUNC8:def_2; A11: (Partial_Sums F) . (k + 1) is_measurable_on E by A4, A6, Th41; A12: F . (k + 1) is nonnegative by A4; A13: (Partial_Sums F) . k is nonnegative by A4, Th36; A14: dom ((Partial_Sums F) . k) = E by A1, A2, A3, Th29; A15: (Partial_Sums F) . k is_measurable_on E by A4, A6, Th41; then consider D being Element of S such that A16: D = dom (((Partial_Sums F) . k) + (F . (k + 1))) and A17: integral+ (M,(((Partial_Sums F) . k) + (F . (k + 1)))) = (integral+ (M,(((Partial_Sums F) . k) | D))) + (integral+ (M,((F . (k + 1)) | D))) by A14, A10, A9, A13, A12, MESFUNC5:78; A18: D = (dom ((Partial_Sums F) . k)) /\ (dom (F . (k + 1))) by A13, A12, A16, MESFUNC5:22 .= E by A14, A10 ; then A19: ((Partial_Sums F) . k) | D = (Partial_Sums F) . k by A14, RELAT_1:68; A20: (F . (k + 1)) | D = F . (k + 1) by A10, A18, RELAT_1:68; dom ((Partial_Sums F) . (k + 1)) = E by A1, A2, A3, Th29; then Integral (M,((Partial_Sums F) . (k + 1))) = integral+ (M,((Partial_Sums F) . (k + 1))) by A4, A11, Th36, MESFUNC5:88 .= (integral+ (M,(((Partial_Sums F) . k) | D))) + (integral+ (M,((F . (k + 1)) | D))) by A17, Def4 .= (Integral (M,((Partial_Sums F) . k))) + (integral+ (M,((F . (k + 1)) | D))) by A14, A15, A13, A19, MESFUNC5:88 .= (Integral (M,((Partial_Sums F) . k))) + (Integral (M,(F . (k + 1)))) by A10, A9, A12, A20, MESFUNC5:88 .= ((Partial_Sums I) . k) + (I . (k + 1)) by A5, A8 ; hence S1[k + 1] by Def1; ::_thesis: verum end; Integral (M,((Partial_Sums F) . 0)) = Integral (M,(F . 0)) by Def4; then Integral (M,((Partial_Sums F) . 0)) = I . 0 by A5; then A21: S1[ 0 ] by Def1; for k being Nat holds S1[k] from NAT_1:sch_2(A21, A7); hence Integral (M,((Partial_Sums F) . n)) = (Partial_Sums I) . n ; ::_thesis: verum end; end; Lm4: for X being non empty set for S being SigmaField of X for M being sigma_Measure of S for E being Element of S for F being Functional_Sequence of X,ExtREAL st E = dom (F . 0) & F is additive & F is with_the_same_dom & ( for n being Nat holds ( F . n is nonnegative & F . n is_measurable_on E ) ) & ( for x being Element of X st x in E holds F # x is summable ) holds ex I being ExtREAL_sequence st ( ( for n being Nat holds I . n = Integral (M,((F . n) | E)) ) & I is summable & Integral (M,((lim (Partial_Sums F)) | E)) = Sum I ) proof let X be non empty set ; ::_thesis: for S being SigmaField of X for M being sigma_Measure of S for E being Element of S for F being Functional_Sequence of X,ExtREAL st E = dom (F . 0) & F is additive & F is with_the_same_dom & ( for n being Nat holds ( F . n is nonnegative & F . n is_measurable_on E ) ) & ( for x being Element of X st x in E holds F # x is summable ) holds ex I being ExtREAL_sequence st ( ( for n being Nat holds I . n = Integral (M,((F . n) | E)) ) & I is summable & Integral (M,((lim (Partial_Sums F)) | E)) = Sum I ) let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S for E being Element of S for F being Functional_Sequence of X,ExtREAL st E = dom (F . 0) & F is additive & F is with_the_same_dom & ( for n being Nat holds ( F . n is nonnegative & F . n is_measurable_on E ) ) & ( for x being Element of X st x in E holds F # x is summable ) holds ex I being ExtREAL_sequence st ( ( for n being Nat holds I . n = Integral (M,((F . n) | E)) ) & I is summable & Integral (M,((lim (Partial_Sums F)) | E)) = Sum I ) let M be sigma_Measure of S; ::_thesis: for E being Element of S for F being Functional_Sequence of X,ExtREAL st E = dom (F . 0) & F is additive & F is with_the_same_dom & ( for n being Nat holds ( F . n is nonnegative & F . n is_measurable_on E ) ) & ( for x being Element of X st x in E holds F # x is summable ) holds ex I being ExtREAL_sequence st ( ( for n being Nat holds I . n = Integral (M,((F . n) | E)) ) & I is summable & Integral (M,((lim (Partial_Sums F)) | E)) = Sum I ) let E be Element of S; ::_thesis: for F being Functional_Sequence of X,ExtREAL st E = dom (F . 0) & F is additive & F is with_the_same_dom & ( for n being Nat holds ( F . n is nonnegative & F . n is_measurable_on E ) ) & ( for x being Element of X st x in E holds F # x is summable ) holds ex I being ExtREAL_sequence st ( ( for n being Nat holds I . n = Integral (M,((F . n) | E)) ) & I is summable & Integral (M,((lim (Partial_Sums F)) | E)) = Sum I ) let F be Functional_Sequence of X,ExtREAL; ::_thesis: ( E = dom (F . 0) & F is additive & F is with_the_same_dom & ( for n being Nat holds ( F . n is nonnegative & F . n is_measurable_on E ) ) & ( for x being Element of X st x in E holds F # x is summable ) implies ex I being ExtREAL_sequence st ( ( for n being Nat holds I . n = Integral (M,((F . n) | E)) ) & I is summable & Integral (M,((lim (Partial_Sums F)) | E)) = Sum I ) ) assume that A1: E = dom (F . 0) and A2: F is additive and A3: F is with_the_same_dom and A4: for n being Nat holds ( F . n is nonnegative & F . n is_measurable_on E ) and A5: for x being Element of X st x in E holds F # x is summable ; ::_thesis: ex I being ExtREAL_sequence st ( ( for n being Nat holds I . n = Integral (M,((F . n) | E)) ) & I is summable & Integral (M,((lim (Partial_Sums F)) | E)) = Sum I ) consider FF being Function of NAT,(Funcs (NAT,(PFuncs (X,ExtREAL)))) such that A6: for n being Nat holds ( ( for m being Nat holds ( (FF . n) . m is_simple_func_in S & dom ((FF . n) . m) = dom (F . n) ) ) & ( for m being Nat holds (FF . n) . m is nonnegative ) & ( for j, k being Nat st j <= k holds for x being Element of X st x in dom (F . n) holds ((FF . n) . j) . x <= ((FF . n) . k) . x ) & ( for x being Element of X st x in dom (F . n) holds ( (FF . n) # x is convergent & lim ((FF . n) # x) = (F . n) . x ) ) ) by A1, A3, A4, Th49; defpred S1[ Element of NAT , Element of NAT , set ] means for n, m being Nat st n = $1 & m = $2 holds $3 = (FF . n) . m; A7: for i1, j1 being Element of NAT ex F1 being Element of PFuncs (X,ExtREAL) st S1[i1,j1,F1] proof let i1, j1 be Element of NAT ; ::_thesis: ex F1 being Element of PFuncs (X,ExtREAL) st S1[i1,j1,F1] reconsider i = i1, j = j1 as Nat ; reconsider F1 = (FF . i) . j as Element of PFuncs (X,ExtREAL) by PARTFUN1:45; take F1 ; ::_thesis: S1[i1,j1,F1] thus S1[i1,j1,F1] ; ::_thesis: verum end; consider FF2 being Function of [:NAT,NAT:],(PFuncs (X,ExtREAL)) such that A8: for i, j being Element of NAT holds S1[i,j,FF2 . (i,j)] from BINOP_1:sch_3(A7); deffunc H1( Nat) -> Element of bool [:X,ExtREAL:] = (Partial_Sums (ProjMap2 (FF2,$1))) . $1; consider P being Functional_Sequence of X,ExtREAL such that A9: for k being Nat holds P . k = H1(k) from SEQFUNC:sch_1(); A10: for n being Nat holds ( ( for m being Nat holds ( dom ((ProjMap1 (FF2,n)) . m) = E & dom ((ProjMap2 (FF2,n)) . m) = E & (ProjMap1 (FF2,n)) . m is_simple_func_in S & (ProjMap2 (FF2,n)) . m is_simple_func_in S ) ) & ProjMap1 (FF2,n) is additive & ProjMap2 (FF2,n) is additive & ProjMap1 (FF2,n) is with_the_same_dom & ProjMap2 (FF2,n) is with_the_same_dom ) proof let n be Nat; ::_thesis: ( ( for m being Nat holds ( dom ((ProjMap1 (FF2,n)) . m) = E & dom ((ProjMap2 (FF2,n)) . m) = E & (ProjMap1 (FF2,n)) . m is_simple_func_in S & (ProjMap2 (FF2,n)) . m is_simple_func_in S ) ) & ProjMap1 (FF2,n) is additive & ProjMap2 (FF2,n) is additive & ProjMap1 (FF2,n) is with_the_same_dom & ProjMap2 (FF2,n) is with_the_same_dom ) A11: now__::_thesis:_for_m_being_Nat_holds_ (_dom_((ProjMap1_(FF2,n))_._m)_=_E_&_dom_((ProjMap2_(FF2,n))_._m)_=_E_&_(ProjMap1_(FF2,n))_._m_is_simple_func_in_S_&_(ProjMap2_(FF2,n))_._m_is_simple_func_in_S_) let m be Nat; ::_thesis: ( dom ((ProjMap1 (FF2,n)) . m) = E & dom ((ProjMap2 (FF2,n)) . m) = E & (ProjMap1 (FF2,n)) . m is_simple_func_in S & (ProjMap2 (FF2,n)) . m is_simple_func_in S ) reconsider n1 = n, m1 = m as Element of NAT by ORDINAL1:def_12; A12: (ProjMap1 (FF2,n)) . m = FF2 . (n,m) by Def8; A13: FF2 . (n1,m1) = (FF . n1) . m by A8; A14: dom (F . m1) = dom (F . 0) by A3, MESFUNC8:def_2; A15: (ProjMap2 (FF2,n)) . m = FF2 . (m,n) by Def9; A16: FF2 . (m1,n1) = (FF . m1) . n by A8; dom (F . n1) = dom (F . m1) by A3, MESFUNC8:def_2; hence ( dom ((ProjMap1 (FF2,n)) . m) = E & dom ((ProjMap2 (FF2,n)) . m) = E ) by A1, A6, A12, A15, A13, A16, A14; ::_thesis: ( (ProjMap1 (FF2,n)) . m is_simple_func_in S & (ProjMap2 (FF2,n)) . m is_simple_func_in S ) thus ( (ProjMap1 (FF2,n)) . m is_simple_func_in S & (ProjMap2 (FF2,n)) . m is_simple_func_in S ) by A6, A12, A15, A13, A16; ::_thesis: verum end; for i1, j1 being Nat holds ( dom ((ProjMap1 (FF2,n)) . i1) = dom ((ProjMap1 (FF2,n)) . j1) & dom ((ProjMap2 (FF2,n)) . i1) = dom ((ProjMap2 (FF2,n)) . j1) ) proof let i1, j1 be Nat; ::_thesis: ( dom ((ProjMap1 (FF2,n)) . i1) = dom ((ProjMap1 (FF2,n)) . j1) & dom ((ProjMap2 (FF2,n)) . i1) = dom ((ProjMap2 (FF2,n)) . j1) ) A17: dom ((ProjMap2 (FF2,n)) . i1) = E by A11; dom ((ProjMap1 (FF2,n)) . i1) = E by A11; hence ( dom ((ProjMap1 (FF2,n)) . i1) = dom ((ProjMap1 (FF2,n)) . j1) & dom ((ProjMap2 (FF2,n)) . i1) = dom ((ProjMap2 (FF2,n)) . j1) ) by A11, A17; ::_thesis: verum end; hence ( ( for m being Nat holds ( dom ((ProjMap1 (FF2,n)) . m) = E & dom ((ProjMap2 (FF2,n)) . m) = E & (ProjMap1 (FF2,n)) . m is_simple_func_in S & (ProjMap2 (FF2,n)) . m is_simple_func_in S ) ) & ProjMap1 (FF2,n) is additive & ProjMap2 (FF2,n) is additive & ProjMap1 (FF2,n) is with_the_same_dom & ProjMap2 (FF2,n) is with_the_same_dom ) by A11, Th35, MESFUNC8:def_2; ::_thesis: verum end; for n, m being Nat holds dom (P . n) = dom (P . m) proof let n, m be Nat; ::_thesis: dom (P . n) = dom (P . m) A18: ProjMap2 (FF2,n) is with_the_same_dom by A10; A19: dom (P . n) = dom ((Partial_Sums (ProjMap2 (FF2,n))) . n) by A9; ProjMap2 (FF2,n) is additive by A10; then dom (P . n) = dom ((ProjMap2 (FF2,n)) . 0) by A18, A19, Th29; then dom (P . n) = E by A10; then A20: dom (P . n) = dom ((ProjMap2 (FF2,m)) . 0) by A10; A21: ProjMap2 (FF2,m) is with_the_same_dom by A10; ProjMap2 (FF2,m) is additive by A10; then dom (P . n) = dom ((Partial_Sums (ProjMap2 (FF2,m))) . m) by A21, A20, Th29; hence dom (P . n) = dom (P . m) by A9; ::_thesis: verum end; then reconsider P = P as with_the_same_dom Functional_Sequence of X,ExtREAL by MESFUNC8:def_2; dom (lim P) = dom (P . 0) by MESFUNC8:def_9; then dom (lim P) = dom ((Partial_Sums (ProjMap2 (FF2,0))) . 0) by A9; then dom (lim P) = dom ((ProjMap2 (FF2,0)) . 0) by Def4; then dom (lim P) = dom (FF2 . (0,0)) by Def9; then A22: dom (lim P) = dom ((FF . 0) . 0) by A8; then A23: dom (lim P) = dom (F . 0) by A6; A24: for k, m being Nat for x being Element of X st x in (dom (F . 0)) /\ (dom ((ProjMap2 (FF2,k)) . 0)) holds ((ProjMap2 (FF2,k)) . m) . x <= (F . m) . x proof let k, m be Nat; ::_thesis: for x being Element of X st x in (dom (F . 0)) /\ (dom ((ProjMap2 (FF2,k)) . 0)) holds ((ProjMap2 (FF2,k)) . m) . x <= (F . m) . x let x be Element of X; ::_thesis: ( x in (dom (F . 0)) /\ (dom ((ProjMap2 (FF2,k)) . 0)) implies ((ProjMap2 (FF2,k)) . m) . x <= (F . m) . x ) reconsider m1 = m, k1 = k as Element of NAT by ORDINAL1:def_12; assume x in (dom (F . 0)) /\ (dom ((ProjMap2 (FF2,k)) . 0)) ; ::_thesis: ((ProjMap2 (FF2,k)) . m) . x <= (F . m) . x then x in dom (F . 0) by XBOOLE_0:def_4; then A25: x in dom (F . m) by A3, MESFUNC8:def_2; (FF . m1) # x is non-decreasing proof let j, k be ext-real number ; :: according to VALUED_0:def_15 ::_thesis: ( not j in dom ((FF . m1) # x) or not k in dom ((FF . m1) # x) or not j <= k or ((FF . m1) # x) . j <= ((FF . m1) # x) . k ) assume that A26: j in dom ((FF . m1) # x) and A27: k in dom ((FF . m1) # x) and A28: j <= k ; ::_thesis: ((FF . m1) # x) . j <= ((FF . m1) # x) . k reconsider j = j, k = k as Element of NAT by A26, A27; A29: ((FF . m1) # x) . k = ((FF . m1) . k) . x by MESFUNC5:def_13; ((FF . m1) # x) . j = ((FF . m1) . j) . x by MESFUNC5:def_13; hence ((FF . m1) # x) . j <= ((FF . m1) # x) . k by A6, A25, A28, A29; ::_thesis: verum end; then lim ((FF . m1) # x) = sup ((FF . m1) # x) by RINFSUP2:37; then ((FF . m1) # x) . k1 <= lim ((FF . m1) # x) by RINFSUP2:23; then A30: ((FF . m1) # x) . k <= (F . m1) . x by A6, A25; (ProjMap2 (FF2,k)) . m = FF2 . (m1,k1) by Def9; then (ProjMap2 (FF2,k)) . m = (FF . m) . k by A8; hence ((ProjMap2 (FF2,k)) . m) . x <= (F . m) . x by A30, MESFUNC5:def_13; ::_thesis: verum end; A31: for x being Element of X for k being Element of NAT st x in dom (lim P) holds (P # x) . k <= ((Partial_Sums F) # x) . k proof let x be Element of X; ::_thesis: for k being Element of NAT st x in dom (lim P) holds (P # x) . k <= ((Partial_Sums F) # x) . k let k be Element of NAT ; ::_thesis: ( x in dom (lim P) implies (P # x) . k <= ((Partial_Sums F) # x) . k ) assume A32: x in dom (lim P) ; ::_thesis: (P # x) . k <= ((Partial_Sums F) # x) . k dom ((ProjMap2 (FF2,k)) . 0) = E by A10; then A33: x in (dom (F . 0)) /\ (dom ((ProjMap2 (FF2,k)) . 0)) by A1, A6, A22, A32; A34: ProjMap2 (FF2,k) is with_the_same_dom by A10; (P # x) . k = (P . k) . x by MESFUNC5:def_13; then A35: (P # x) . k = ((Partial_Sums (ProjMap2 (FF2,k))) . k) . x by A9; A36: for m being Nat for x being Element of X st x in (dom (F . 0)) /\ (dom ((ProjMap2 (FF2,k)) . 0)) holds ((ProjMap2 (FF2,k)) . m) . x <= (F . m) . x by A24; ProjMap2 (FF2,k) is additive by A10; then ((Partial_Sums (ProjMap2 (FF2,k))) . k) . x <= ((Partial_Sums F) . k) . x by A2, A3, A33, A34, A36, Th42; hence (P # x) . k <= ((Partial_Sums F) # x) . k by A35, MESFUNC5:def_13; ::_thesis: verum end; dom (lim (Partial_Sums F)) = dom ((Partial_Sums F) . 0) by MESFUNC8:def_9; then A37: dom (lim (Partial_Sums F)) = E by A1, Def4; A38: for n, m being Nat holds ( (ProjMap1 (FF2,n)) . m is nonnegative & (ProjMap2 (FF2,n)) . m is nonnegative ) proof let n, m be Nat; ::_thesis: ( (ProjMap1 (FF2,n)) . m is nonnegative & (ProjMap2 (FF2,n)) . m is nonnegative ) reconsider n1 = n, m1 = m as Element of NAT by ORDINAL1:def_12; (ProjMap1 (FF2,n)) . m = FF2 . (n1,m1) by Def8; then (ProjMap1 (FF2,n)) . m = (FF . n) . m by A8; hence (ProjMap1 (FF2,n)) . m is nonnegative by A6; ::_thesis: (ProjMap2 (FF2,n)) . m is nonnegative (ProjMap2 (FF2,n)) . m = FF2 . (m1,n1) by Def9; then (ProjMap2 (FF2,n)) . m = (FF . m) . n by A8; hence (ProjMap2 (FF2,n)) . m is nonnegative by A6; ::_thesis: verum end; A39: for n being Element of NAT for x being Element of X st x in E holds ( (ProjMap1 (FF2,n)) # x is convergent & (F . n) . x = lim ((ProjMap1 (FF2,n)) # x) ) proof let n be Element of NAT ; ::_thesis: for x being Element of X st x in E holds ( (ProjMap1 (FF2,n)) # x is convergent & (F . n) . x = lim ((ProjMap1 (FF2,n)) # x) ) reconsider n1 = n as Nat ; let x be Element of X; ::_thesis: ( x in E implies ( (ProjMap1 (FF2,n)) # x is convergent & (F . n) . x = lim ((ProjMap1 (FF2,n)) # x) ) ) assume A40: x in E ; ::_thesis: ( (ProjMap1 (FF2,n)) # x is convergent & (F . n) . x = lim ((ProjMap1 (FF2,n)) # x) ) for k being Nat ex m being Nat st ( k <= m & ((ProjMap1 (FF2,n)) # x) . m > - 1 ) proof let k be Nat; ::_thesis: ex m being Nat st ( k <= m & ((ProjMap1 (FF2,n)) # x) . m > - 1 ) take m = k; ::_thesis: ( k <= m & ((ProjMap1 (FF2,n)) # x) . m > - 1 ) A41: ((ProjMap1 (FF2,n)) # x) . m = ((ProjMap1 (FF2,n)) . m) . x by MESFUNC5:def_13; (ProjMap1 (FF2,n)) . m is nonnegative by A38; hence ( k <= m & ((ProjMap1 (FF2,n)) # x) . m > - 1 ) by A41, SUPINF_2:39; ::_thesis: verum end; then A42: not (ProjMap1 (FF2,n)) # x is convergent_to_-infty by MESFUNC5:def_10; A43: E = dom (F . n1) by A1, A3, MESFUNC8:def_2; (ProjMap1 (FF2,n)) # x is non-decreasing proof let i, j be ext-real number ; :: according to VALUED_0:def_15 ::_thesis: ( not i in dom ((ProjMap1 (FF2,n)) # x) or not j in dom ((ProjMap1 (FF2,n)) # x) or not i <= j or ((ProjMap1 (FF2,n)) # x) . i <= ((ProjMap1 (FF2,n)) # x) . j ) assume that A44: i in dom ((ProjMap1 (FF2,n)) # x) and A45: j in dom ((ProjMap1 (FF2,n)) # x) and A46: i <= j ; ::_thesis: ((ProjMap1 (FF2,n)) # x) . i <= ((ProjMap1 (FF2,n)) # x) . j reconsider i1 = i, j1 = j as Element of NAT by A44, A45; A47: ((ProjMap1 (FF2,n)) # x) . i1 = ((ProjMap1 (FF2,n)) . i1) . x by MESFUNC5:def_13; (ProjMap1 (FF2,n)) . i1 = FF2 . (n,i1) by Def8; then A48: (ProjMap1 (FF2,n)) . i1 = (FF . n1) . i1 by A8; A49: ((ProjMap1 (FF2,n)) # x) . j1 = ((ProjMap1 (FF2,n)) . j1) . x by MESFUNC5:def_13; A50: (ProjMap1 (FF2,n)) . j1 = FF2 . (n,j1) by Def8; ((FF . n1) . i1) . x <= ((FF . n1) . j1) . x by A6, A43, A40, A46; hence ((ProjMap1 (FF2,n)) # x) . i <= ((ProjMap1 (FF2,n)) # x) . j by A8, A47, A49, A48, A50; ::_thesis: verum end; hence A51: (ProjMap1 (FF2,n)) # x is convergent by RINFSUP2:37; ::_thesis: (F . n) . x = lim ((ProjMap1 (FF2,n)) # x) percases ( (ProjMap1 (FF2,n)) # x is convergent_to_finite_number or (ProjMap1 (FF2,n)) # x is convergent_to_+infty ) by A51, A42, MESFUNC5:def_11; supposeA52: (ProjMap1 (FF2,n)) # x is convergent_to_finite_number ; ::_thesis: (F . n) . x = lim ((ProjMap1 (FF2,n)) # x) then A53: not (ProjMap1 (FF2,n)) # x is convergent_to_-infty by MESFUNC5:51; not (ProjMap1 (FF2,n)) # x is convergent_to_+infty by A52, MESFUNC5:50; then consider lP being real number such that A54: lim ((ProjMap1 (FF2,n)) # x) = lP and A55: for p being real number st 0 < p holds ex nn being Nat st for mm being Nat st nn <= mm holds |.((((ProjMap1 (FF2,n)) # x) . mm) - (lim ((ProjMap1 (FF2,n)) # x))).| < p and (ProjMap1 (FF2,n)) # x is convergent_to_finite_number by A51, A53, MESFUNC5:def_12; A56: for p being real number st 0 < p holds ex nn being Nat st for mm being Nat st nn <= mm holds |.((((FF . n1) # x) . mm) - (R_EAL lP)).| < p proof let p be real number ; ::_thesis: ( 0 < p implies ex nn being Nat st for mm being Nat st nn <= mm holds |.((((FF . n1) # x) . mm) - (R_EAL lP)).| < p ) assume 0 < p ; ::_thesis: ex nn being Nat st for mm being Nat st nn <= mm holds |.((((FF . n1) # x) . mm) - (R_EAL lP)).| < p then consider nn being Nat such that A57: for mm being Nat st nn <= mm holds |.((((ProjMap1 (FF2,n)) # x) . mm) - (lim ((ProjMap1 (FF2,n)) # x))).| < p by A55; take nn ; ::_thesis: for mm being Nat st nn <= mm holds |.((((FF . n1) # x) . mm) - (R_EAL lP)).| < p let mm be Nat; ::_thesis: ( nn <= mm implies |.((((FF . n1) # x) . mm) - (R_EAL lP)).| < p ) assume A58: nn <= mm ; ::_thesis: |.((((FF . n1) # x) . mm) - (R_EAL lP)).| < p reconsider mm1 = mm as Element of NAT by ORDINAL1:def_12; (ProjMap1 (FF2,n)) . mm = FF2 . (n,mm) by Def8; then A59: (ProjMap1 (FF2,n)) . mm = (FF . n1) . mm1 by A8; ((ProjMap1 (FF2,n)) # x) . mm = ((ProjMap1 (FF2,n)) . mm) . x by MESFUNC5:def_13; then ((FF . n1) # x) . mm = ((ProjMap1 (FF2,n)) # x) . mm by A59, MESFUNC5:def_13; hence |.((((FF . n1) # x) . mm) - (R_EAL lP)).| < p by A54, A57, A58; ::_thesis: verum end; then A60: (FF . n1) # x is convergent_to_finite_number by MESFUNC5:def_8; then (FF . n1) # x is convergent by MESFUNC5:def_11; then lim ((FF . n1) # x) = R_EAL lP by A56, A60, MESFUNC5:def_12; hence (F . n) . x = lim ((ProjMap1 (FF2,n)) # x) by A6, A43, A40, A54; ::_thesis: verum end; supposeA61: (ProjMap1 (FF2,n)) # x is convergent_to_+infty ; ::_thesis: (F . n) . x = lim ((ProjMap1 (FF2,n)) # x) for g being real number st 0 < g holds ex nn being Nat st for mm being Nat st nn <= mm holds g <= ((FF . n1) # x) . mm proof let g be real number ; ::_thesis: ( 0 < g implies ex nn being Nat st for mm being Nat st nn <= mm holds g <= ((FF . n1) # x) . mm ) assume 0 < g ; ::_thesis: ex nn being Nat st for mm being Nat st nn <= mm holds g <= ((FF . n1) # x) . mm then consider nn being Nat such that A62: for mm being Nat st nn <= mm holds g <= ((ProjMap1 (FF2,n)) # x) . mm by A61, MESFUNC5:def_9; take nn ; ::_thesis: for mm being Nat st nn <= mm holds g <= ((FF . n1) # x) . mm let mm be Nat; ::_thesis: ( nn <= mm implies g <= ((FF . n1) # x) . mm ) assume nn <= mm ; ::_thesis: g <= ((FF . n1) # x) . mm then A63: g <= ((ProjMap1 (FF2,n)) # x) . mm by A62; reconsider mm1 = mm as Element of NAT by ORDINAL1:def_12; (ProjMap1 (FF2,n)) . mm = FF2 . (n,mm1) by Def8; then A64: (ProjMap1 (FF2,n)) . mm = (FF . n) . mm by A8; ((ProjMap1 (FF2,n)) # x) . mm = ((ProjMap1 (FF2,n)) . mm) . x by MESFUNC5:def_13; hence g <= ((FF . n1) # x) . mm by A63, A64, MESFUNC5:def_13; ::_thesis: verum end; then A65: (FF . n1) # x is convergent_to_+infty by MESFUNC5:def_9; then (FF . n1) # x is convergent by MESFUNC5:def_11; then A66: lim ((FF . n1) # x) = +infty by A65, MESFUNC5:def_12; lim ((ProjMap1 (FF2,n)) # x) = +infty by A51, A61, MESFUNC5:def_12; hence (F . n) . x = lim ((ProjMap1 (FF2,n)) # x) by A6, A43, A40, A66; ::_thesis: verum end; end; end; A67: dom (lim (Partial_Sums F)) = dom ((Partial_Sums F) . 0) by MESFUNC8:def_9; then A68: dom (lim (Partial_Sums F)) = E by A1, Def4; A69: for n being Nat holds dom (P . n) = dom (lim (Partial_Sums F)) proof let n be Nat; ::_thesis: dom (P . n) = dom (lim (Partial_Sums F)) A70: ProjMap2 (FF2,n) is with_the_same_dom by A10; A71: dom (P . n) = dom ((Partial_Sums (ProjMap2 (FF2,n))) . n) by A9; ProjMap2 (FF2,n) is additive by A10; then dom (P . n) = dom ((ProjMap2 (FF2,n)) . 0) by A70, A71, Th29; hence dom (P . n) = dom (lim (Partial_Sums F)) by A68, A10; ::_thesis: verum end; A72: for n, m being Nat st n <= m holds for i being Nat for x being Element of X st x in E holds ((ProjMap2 (FF2,n)) . i) . x <= ((ProjMap2 (FF2,m)) . i) . x proof let n, m be Nat; ::_thesis: ( n <= m implies for i being Nat for x being Element of X st x in E holds ((ProjMap2 (FF2,n)) . i) . x <= ((ProjMap2 (FF2,m)) . i) . x ) assume A73: n <= m ; ::_thesis: for i being Nat for x being Element of X st x in E holds ((ProjMap2 (FF2,n)) . i) . x <= ((ProjMap2 (FF2,m)) . i) . x let i be Nat; ::_thesis: for x being Element of X st x in E holds ((ProjMap2 (FF2,n)) . i) . x <= ((ProjMap2 (FF2,m)) . i) . x let x be Element of X; ::_thesis: ( x in E implies ((ProjMap2 (FF2,n)) . i) . x <= ((ProjMap2 (FF2,m)) . i) . x ) reconsider i1 = i, n1 = n, m1 = m as Element of NAT by ORDINAL1:def_12; ((ProjMap2 (FF2,n)) . i) . x = (FF2 . (i1,n1)) . x by Def9; then A74: ((ProjMap2 (FF2,n)) . i) . x = ((FF . i) . n) . x by A8; ((ProjMap2 (FF2,m)) . i) . x = (FF2 . (i1,m1)) . x by Def9; then A75: ((ProjMap2 (FF2,m)) . i) . x = ((FF . i) . m) . x by A8; assume x in E ; ::_thesis: ((ProjMap2 (FF2,n)) . i) . x <= ((ProjMap2 (FF2,m)) . i) . x then x in dom (F . i) by A1, A3, MESFUNC8:def_2; hence ((ProjMap2 (FF2,n)) . i) . x <= ((ProjMap2 (FF2,m)) . i) . x by A6, A73, A74, A75; ::_thesis: verum end; A76: for n, m being Nat st n <= m holds for i being Nat for x being Element of X st x in E holds ((Partial_Sums (ProjMap2 (FF2,n))) . i) . x <= ((Partial_Sums (ProjMap2 (FF2,m))) . i) . x proof let n, m be Nat; ::_thesis: ( n <= m implies for i being Nat for x being Element of X st x in E holds ((Partial_Sums (ProjMap2 (FF2,n))) . i) . x <= ((Partial_Sums (ProjMap2 (FF2,m))) . i) . x ) A77: ProjMap2 (FF2,n) is with_the_same_dom by A10; A78: ProjMap2 (FF2,m) is additive by A10; assume A79: n <= m ; ::_thesis: for i being Nat for x being Element of X st x in E holds ((Partial_Sums (ProjMap2 (FF2,n))) . i) . x <= ((Partial_Sums (ProjMap2 (FF2,m))) . i) . x A80: for i being Nat for x being Element of X st x in (dom ((ProjMap2 (FF2,n)) . 0)) /\ (dom ((ProjMap2 (FF2,m)) . 0)) holds ((ProjMap2 (FF2,n)) . i) . x <= ((ProjMap2 (FF2,m)) . i) . x proof let i be Nat; ::_thesis: for x being Element of X st x in (dom ((ProjMap2 (FF2,n)) . 0)) /\ (dom ((ProjMap2 (FF2,m)) . 0)) holds ((ProjMap2 (FF2,n)) . i) . x <= ((ProjMap2 (FF2,m)) . i) . x let x be Element of X; ::_thesis: ( x in (dom ((ProjMap2 (FF2,n)) . 0)) /\ (dom ((ProjMap2 (FF2,m)) . 0)) implies ((ProjMap2 (FF2,n)) . i) . x <= ((ProjMap2 (FF2,m)) . i) . x ) assume x in (dom ((ProjMap2 (FF2,n)) . 0)) /\ (dom ((ProjMap2 (FF2,m)) . 0)) ; ::_thesis: ((ProjMap2 (FF2,n)) . i) . x <= ((ProjMap2 (FF2,m)) . i) . x then x in dom ((ProjMap2 (FF2,n)) . 0) by XBOOLE_0:def_4; then x in E by A10; hence ((ProjMap2 (FF2,n)) . i) . x <= ((ProjMap2 (FF2,m)) . i) . x by A72, A79; ::_thesis: verum end; let i be Nat; ::_thesis: for x being Element of X st x in E holds ((Partial_Sums (ProjMap2 (FF2,n))) . i) . x <= ((Partial_Sums (ProjMap2 (FF2,m))) . i) . x let x be Element of X; ::_thesis: ( x in E implies ((Partial_Sums (ProjMap2 (FF2,n))) . i) . x <= ((Partial_Sums (ProjMap2 (FF2,m))) . i) . x ) assume A81: x in E ; ::_thesis: ((Partial_Sums (ProjMap2 (FF2,n))) . i) . x <= ((Partial_Sums (ProjMap2 (FF2,m))) . i) . x then A82: x in dom ((ProjMap2 (FF2,m)) . 0) by A10; x in dom ((ProjMap2 (FF2,n)) . 0) by A10, A81; then A83: x in (dom ((ProjMap2 (FF2,n)) . 0)) /\ (dom ((ProjMap2 (FF2,m)) . 0)) by A82, XBOOLE_0:def_4; A84: ProjMap2 (FF2,m) is with_the_same_dom by A10; ProjMap2 (FF2,n) is additive by A10; hence ((Partial_Sums (ProjMap2 (FF2,n))) . i) . x <= ((Partial_Sums (ProjMap2 (FF2,m))) . i) . x by A83, A77, A78, A84, A80, Th42; ::_thesis: verum end; A85: for n, m being Nat st n <= m holds for x being Element of X st x in E holds (P . n) . x <= (P . m) . x proof let n, m be Nat; ::_thesis: ( n <= m implies for x being Element of X st x in E holds (P . n) . x <= (P . m) . x ) reconsider n1 = n, m1 = m as Element of NAT by ORDINAL1:def_12; assume A86: n <= m ; ::_thesis: for x being Element of X st x in E holds (P . n) . x <= (P . m) . x let x be Element of X; ::_thesis: ( x in E implies (P . n) . x <= (P . m) . x ) A87: ProjMap2 (FF2,m) is with_the_same_dom by A10; A88: for n being Nat holds (ProjMap2 (FF2,m)) . n is nonnegative by A38; assume A89: x in E ; ::_thesis: (P . n) . x <= (P . m) . x then x in dom ((ProjMap2 (FF2,m)) . 0) by A10; then (Partial_Sums (ProjMap2 (FF2,m))) # x is non-decreasing by A87, A88, Th38; then ((Partial_Sums (ProjMap2 (FF2,m))) # x) . n1 <= ((Partial_Sums (ProjMap2 (FF2,m))) # x) . m1 by A86, RINFSUP2:7; then ((Partial_Sums (ProjMap2 (FF2,m))) . n) . x <= ((Partial_Sums (ProjMap2 (FF2,m))) # x) . m1 by MESFUNC5:def_13; then ((Partial_Sums (ProjMap2 (FF2,m))) . n) . x <= ((Partial_Sums (ProjMap2 (FF2,m))) . m) . x by MESFUNC5:def_13; then A90: ((Partial_Sums (ProjMap2 (FF2,m))) . n) . x <= (P . m) . x by A9; ((Partial_Sums (ProjMap2 (FF2,n))) . n) . x <= ((Partial_Sums (ProjMap2 (FF2,m))) . n) . x by A76, A86, A89; then (P . n) . x <= ((Partial_Sums (ProjMap2 (FF2,m))) . n) . x by A9; hence (P . n) . x <= (P . m) . x by A90, XXREAL_0:2; ::_thesis: verum end; A91: for x being Element of X st x in dom (lim P) holds P # x is convergent proof let x be Element of X; ::_thesis: ( x in dom (lim P) implies P # x is convergent ) assume A92: x in dom (lim P) ; ::_thesis: P # x is convergent for n, m being Element of NAT st m <= n holds (P # x) . m <= (P # x) . n proof let n, m be Element of NAT ; ::_thesis: ( m <= n implies (P # x) . m <= (P # x) . n ) assume m <= n ; ::_thesis: (P # x) . m <= (P # x) . n then (P . m) . x <= (P . n) . x by A1, A85, A23, A92; then (P # x) . m <= (P . n) . x by MESFUNC5:def_13; hence (P # x) . m <= (P # x) . n by MESFUNC5:def_13; ::_thesis: verum end; then P # x is non-decreasing by RINFSUP2:7; hence P # x is convergent by RINFSUP2:37; ::_thesis: verum end; A93: for x being Element of X st x in dom (lim P) holds lim (P # x) = lim ((Partial_Sums F) # x) proof defpred S2[ Element of NAT , Element of NAT , set ] means for n, m being Nat st n = $1 & m = $2 holds $3 = (Partial_Sums (ProjMap2 (FF2,n))) . m; let x be Element of X; ::_thesis: ( x in dom (lim P) implies lim (P # x) = lim ((Partial_Sums F) # x) ) A94: for i1, j1 being Element of NAT ex PP2 being Element of PFuncs (X,ExtREAL) st S2[i1,j1,PP2] proof let i1, j1 be Element of NAT ; ::_thesis: ex PP2 being Element of PFuncs (X,ExtREAL) st S2[i1,j1,PP2] reconsider i = i1, j = j1 as Nat ; reconsider F1 = (Partial_Sums (ProjMap2 (FF2,i))) . j as Element of PFuncs (X,ExtREAL) by PARTFUN1:45; take F1 ; ::_thesis: S2[i1,j1,F1] thus S2[i1,j1,F1] ; ::_thesis: verum end; consider PP2 being Function of [:NAT,NAT:],(PFuncs (X,ExtREAL)) such that A95: for i, j being Element of NAT holds S2[i,j,PP2 . (i,j)] from BINOP_1:sch_3(A94); assume A96: x in dom (lim P) ; ::_thesis: lim (P # x) = lim ((Partial_Sums F) # x) then A97: P # x is convergent by A91; A98: for p, n being Element of NAT holds (ProjMap2 (PP2,n)) . p = (Partial_Sums (ProjMap2 (FF2,p))) . n proof let p, n be Element of NAT ; ::_thesis: (ProjMap2 (PP2,n)) . p = (Partial_Sums (ProjMap2 (FF2,p))) . n (ProjMap2 (PP2,n)) . p = PP2 . (p,n) by Def9; hence (ProjMap2 (PP2,n)) . p = (Partial_Sums (ProjMap2 (FF2,p))) . n by A95; ::_thesis: verum end; A99: for n being Element of NAT holds ( (ProjMap2 (PP2,n)) # x is convergent & ((ProjMap2 (PP2,n)) # x) ^\ n is convergent & lim ((ProjMap2 (PP2,n)) # x) = lim (((ProjMap2 (PP2,n)) # x) ^\ n) ) proof let n be Element of NAT ; ::_thesis: ( (ProjMap2 (PP2,n)) # x is convergent & ((ProjMap2 (PP2,n)) # x) ^\ n is convergent & lim ((ProjMap2 (PP2,n)) # x) = lim (((ProjMap2 (PP2,n)) # x) ^\ n) ) (ProjMap2 (PP2,n)) # x is non-decreasing proof let j, k be ext-real number ; :: according to VALUED_0:def_15 ::_thesis: ( not j in dom ((ProjMap2 (PP2,n)) # x) or not k in dom ((ProjMap2 (PP2,n)) # x) or not j <= k or ((ProjMap2 (PP2,n)) # x) . j <= ((ProjMap2 (PP2,n)) # x) . k ) assume that A100: j in dom ((ProjMap2 (PP2,n)) # x) and A101: k in dom ((ProjMap2 (PP2,n)) # x) and A102: j <= k ; ::_thesis: ((ProjMap2 (PP2,n)) # x) . j <= ((ProjMap2 (PP2,n)) # x) . k reconsider j = j, k = k as Element of NAT by A100, A101; A103: ProjMap2 (FF2,j) is additive by A10; A104: ProjMap2 (FF2,k) is with_the_same_dom by A10; A105: dom ((ProjMap2 (FF2,k)) . 0) = E by A10; ((ProjMap2 (PP2,n)) # x) . k = ((ProjMap2 (PP2,n)) . k) . x by MESFUNC5:def_13; then A106: ((ProjMap2 (PP2,n)) # x) . k = ((Partial_Sums (ProjMap2 (FF2,k))) . n) . x by A98; A107: ProjMap2 (FF2,k) is additive by A10; ((ProjMap2 (PP2,n)) # x) . j = ((ProjMap2 (PP2,n)) . j) . x by MESFUNC5:def_13; then A108: ((ProjMap2 (PP2,n)) # x) . j = ((Partial_Sums (ProjMap2 (FF2,j))) . n) . x by A98; A109: ProjMap2 (FF2,j) is with_the_same_dom by A10; A110: dom ((ProjMap2 (FF2,j)) . 0) = E by A10; then for i being Nat for z being Element of X st z in (dom ((ProjMap2 (FF2,j)) . 0)) /\ (dom ((ProjMap2 (FF2,k)) . 0)) holds ((ProjMap2 (FF2,j)) . i) . z <= ((ProjMap2 (FF2,k)) . i) . z by A72, A102, A105; hence ((ProjMap2 (PP2,n)) # x) . j <= ((ProjMap2 (PP2,n)) # x) . k by A1, A23, A96, A108, A106, A103, A109, A107, A104, A110, A105, Th42; ::_thesis: verum end; hence (ProjMap2 (PP2,n)) # x is convergent by RINFSUP2:37; ::_thesis: ( ((ProjMap2 (PP2,n)) # x) ^\ n is convergent & lim ((ProjMap2 (PP2,n)) # x) = lim (((ProjMap2 (PP2,n)) # x) ^\ n) ) hence ( ((ProjMap2 (PP2,n)) # x) ^\ n is convergent & lim ((ProjMap2 (PP2,n)) # x) = lim (((ProjMap2 (PP2,n)) # x) ^\ n) ) by RINFSUP2:21; ::_thesis: verum end; A111: for n being Nat holds ((Partial_Sums F) # x) . n <= lim (P # x) proof for p being set st p in NAT holds ((ProjMap2 (PP2,0)) # x) . p = ((ProjMap1 (FF2,0)) # x) . p proof let p be set ; ::_thesis: ( p in NAT implies ((ProjMap2 (PP2,0)) # x) . p = ((ProjMap1 (FF2,0)) # x) . p ) assume p in NAT ; ::_thesis: ((ProjMap2 (PP2,0)) # x) . p = ((ProjMap1 (FF2,0)) # x) . p then reconsider p9 = p as Element of NAT ; (ProjMap2 (PP2,0)) . p9 = (Partial_Sums (ProjMap2 (FF2,p9))) . 0 by A98; then (ProjMap2 (PP2,0)) . p9 = (ProjMap2 (FF2,p9)) . 0 by Def4; then (ProjMap2 (PP2,0)) . p9 = FF2 . (0,p9) by Def9; then A112: (ProjMap2 (PP2,0)) . p9 = (ProjMap1 (FF2,0)) . p9 by Def8; ((ProjMap2 (PP2,0)) # x) . p = ((ProjMap2 (PP2,0)) . p9) . x by MESFUNC5:def_13; hence ((ProjMap2 (PP2,0)) # x) . p = ((ProjMap1 (FF2,0)) # x) . p by A112, MESFUNC5:def_13; ::_thesis: verum end; then (ProjMap2 (PP2,0)) # x = (ProjMap1 (FF2,0)) # x by FUNCT_2:12; then A113: lim ((ProjMap2 (PP2,0)) # x) = (F . 0) . x by A1, A39, A23, A96; defpred S3[ Nat] means lim ((ProjMap2 (PP2,$1)) # x) = ((Partial_Sums F) # x) . $1; let n be Nat; ::_thesis: ((Partial_Sums F) # x) . n <= lim (P # x) reconsider n9 = n as Element of NAT by ORDINAL1:def_12; A114: lim ((P # x) ^\ n9) = lim (P # x) by A97, RINFSUP2:21; A115: ((ProjMap2 (PP2,n)) # x) ^\ n9 is convergent by A99; A116: for k being Nat st S3[k] holds S3[k + 1] proof let k be Nat; ::_thesis: ( S3[k] implies S3[k + 1] ) reconsider k9 = k as Element of NAT by ORDINAL1:def_12; assume A117: S3[k] ; ::_thesis: S3[k + 1] A118: (ProjMap2 (PP2,k9)) # x is convergent by A99; now__::_thesis:_for_m_being_set_st_m_in_dom_((ProjMap1_(FF2,(k_+_1)))_#_x)_holds_ 0._<=_((ProjMap1_(FF2,(k_+_1)))_#_x)_._m let m be set ; ::_thesis: ( m in dom ((ProjMap1 (FF2,(k + 1))) # x) implies 0. <= ((ProjMap1 (FF2,(k + 1))) # x) . m ) assume m in dom ((ProjMap1 (FF2,(k + 1))) # x) ; ::_thesis: 0. <= ((ProjMap1 (FF2,(k + 1))) # x) . m then reconsider m1 = m as Element of NAT ; (ProjMap1 (FF2,(k + 1))) . m1 is nonnegative by A38; then 0. <= ((ProjMap1 (FF2,(k + 1))) . m1) . x by SUPINF_2:51; hence 0. <= ((ProjMap1 (FF2,(k + 1))) # x) . m by MESFUNC5:def_13; ::_thesis: verum end; then A119: (ProjMap1 (FF2,(k + 1))) # x is V111() by SUPINF_2:52; now__::_thesis:_for_m_being_set_st_m_in_dom_((ProjMap2_(PP2,k))_#_x)_holds_ 0._<=_((ProjMap2_(PP2,k))_#_x)_._m let m be set ; ::_thesis: ( m in dom ((ProjMap2 (PP2,k)) # x) implies 0. <= ((ProjMap2 (PP2,k)) # x) . m ) assume m in dom ((ProjMap2 (PP2,k)) # x) ; ::_thesis: 0. <= ((ProjMap2 (PP2,k)) # x) . m then reconsider m1 = m as Element of NAT ; A120: (ProjMap2 (PP2,k)) . m1 = (Partial_Sums (ProjMap2 (FF2,m1))) . k9 by A98; for l being Nat holds (ProjMap2 (FF2,m1)) . l is nonnegative by A38; then (ProjMap2 (PP2,k)) . m1 is nonnegative by A120, Th36; then 0. <= ((ProjMap2 (PP2,k)) . m1) . x by SUPINF_2:51; hence 0. <= ((ProjMap2 (PP2,k)) # x) . m by MESFUNC5:def_13; ::_thesis: verum end; then A121: (ProjMap2 (PP2,k)) # x is V111() by SUPINF_2:52; x in dom ((Partial_Sums F) . (k + 1)) by A2, A3, A23, A96, Th29; then A122: x in dom (((Partial_Sums F) . k) + (F . (k + 1))) by Def4; A123: for p being Nat holds ((ProjMap2 (PP2,(k + 1))) # x) . p = (((ProjMap2 (PP2,k)) # x) . p) + (((ProjMap1 (FF2,(k + 1))) # x) . p) proof let p be Nat; ::_thesis: ((ProjMap2 (PP2,(k + 1))) # x) . p = (((ProjMap2 (PP2,k)) # x) . p) + (((ProjMap1 (FF2,(k + 1))) # x) . p) reconsider p9 = p as Element of NAT by ORDINAL1:def_12; A124: (ProjMap2 (FF2,p9)) . (k + 1) = FF2 . ((k + 1),p9) by Def9; A125: ProjMap2 (FF2,p) is with_the_same_dom by A10; A126: dom ((ProjMap2 (FF2,p)) . 0) = E by A10; ProjMap2 (FF2,p) is additive by A10; then E c= dom ((Partial_Sums (ProjMap2 (FF2,p))) . (k + 1)) by A125, A126, Th29; then A127: E c= dom ((ProjMap2 (PP2,(k + 1))) . p9) by A98; (ProjMap2 (PP2,(k + 1))) . p9 = (Partial_Sums (ProjMap2 (FF2,p9))) . (k + 1) by A98; then A128: (ProjMap2 (PP2,(k + 1))) . p9 = ((Partial_Sums (ProjMap2 (FF2,p9))) . k) + ((ProjMap2 (FF2,p9)) . (k + 1)) by Def4; (Partial_Sums (ProjMap2 (FF2,p9))) . k9 = (ProjMap2 (PP2,k)) . p9 by A98; then (ProjMap2 (PP2,(k + 1))) . p9 = ((ProjMap2 (PP2,k)) . p9) + ((ProjMap1 (FF2,(k + 1))) . p9) by A128, A124, Def8; then ((ProjMap2 (PP2,(k + 1))) . p9) . x = (((ProjMap2 (PP2,k)) . p9) . x) + (((ProjMap1 (FF2,(k + 1))) . p9) . x) by A1, A23, A96, A127, MESFUNC1:def_3; then A129: ((ProjMap2 (PP2,(k + 1))) . p9) . x = (((ProjMap2 (PP2,k)) # x) . p) + (((ProjMap1 (FF2,(k + 1))) . p9) . x) by MESFUNC5:def_13; ((ProjMap2 (PP2,(k + 1))) # x) . p = ((ProjMap2 (PP2,(k + 1))) . p9) . x by MESFUNC5:def_13; hence ((ProjMap2 (PP2,(k + 1))) # x) . p = (((ProjMap2 (PP2,k)) # x) . p) + (((ProjMap1 (FF2,(k + 1))) # x) . p) by A129, MESFUNC5:def_13; ::_thesis: verum end; A130: lim ((ProjMap1 (FF2,(k + 1))) # x) = (F . (k + 1)) . x by A1, A39, A23, A96; (ProjMap1 (FF2,(k + 1))) # x is convergent by A1, A39, A23, A96; then lim ((ProjMap2 (PP2,(k + 1))) # x) = (lim ((ProjMap2 (PP2,k)) # x)) + (lim ((ProjMap1 (FF2,(k + 1))) # x)) by A118, A121, A119, A123, Th11; then lim ((ProjMap2 (PP2,(k + 1))) # x) = (((Partial_Sums F) . k) . x) + ((F . (k + 1)) . x) by A117, A130, MESFUNC5:def_13; then lim ((ProjMap2 (PP2,(k + 1))) # x) = (((Partial_Sums F) . k) + (F . (k + 1))) . x by A122, MESFUNC1:def_3; then lim ((ProjMap2 (PP2,(k + 1))) # x) = ((Partial_Sums F) . (k + 1)) . x by Def4; hence S3[k + 1] by MESFUNC5:def_13; ::_thesis: verum end; A131: for p being Element of NAT holds (((ProjMap2 (PP2,n)) # x) ^\ n9) . p <= ((P # x) ^\ n9) . p proof let p be Element of NAT ; ::_thesis: (((ProjMap2 (PP2,n)) # x) ^\ n9) . p <= ((P # x) ^\ n9) . p A132: n <= n + p by NAT_1:11; A133: ProjMap2 (FF2,(n + p)) is with_the_same_dom by A10; A134: for i being Nat holds (ProjMap2 (FF2,(n + p))) . i is nonnegative by A38; x in dom ((ProjMap2 (FF2,(n + p))) . 0) by A1, A10, A23, A96; then (Partial_Sums (ProjMap2 (FF2,(n + p)))) # x is non-decreasing by A133, A134, Th38; then ((Partial_Sums (ProjMap2 (FF2,(n + p)))) # x) . n9 <= ((Partial_Sums (ProjMap2 (FF2,(n + p)))) # x) . (n9 + p) by A132, RINFSUP2:7; then A135: ((Partial_Sums (ProjMap2 (FF2,(n + p)))) # x) . n9 <= ((Partial_Sums (ProjMap2 (FF2,(n + p)))) . (n + p)) . x by MESFUNC5:def_13; ((P # x) ^\ n9) . p = (P # x) . (n + p) by NAT_1:def_3; then ((P # x) ^\ n9) . p = (P . (n + p)) . x by MESFUNC5:def_13; then A136: ((P # x) ^\ n9) . p = ((Partial_Sums (ProjMap2 (FF2,(n + p)))) . (n + p)) . x by A9; (((ProjMap2 (PP2,n)) # x) ^\ n9) . p = ((ProjMap2 (PP2,n)) # x) . (n + p) by NAT_1:def_3; then (((ProjMap2 (PP2,n)) # x) ^\ n9) . p = ((ProjMap2 (PP2,n)) . (n + p)) . x by MESFUNC5:def_13; then (((ProjMap2 (PP2,n)) # x) ^\ n9) . p = ((Partial_Sums (ProjMap2 (FF2,(n + p)))) . n) . x by A98; hence (((ProjMap2 (PP2,n)) # x) ^\ n9) . p <= ((P # x) ^\ n9) . p by A136, A135, MESFUNC5:def_13; ::_thesis: verum end; ((Partial_Sums F) # x) . 0 = ((Partial_Sums F) . 0) . x by MESFUNC5:def_13; then A137: S3[ 0 ] by A113, Def4; A138: for k being Nat holds S3[k] from NAT_1:sch_2(A137, A116); (P # x) ^\ n9 is convergent by A97, RINFSUP2:21; then lim (((ProjMap2 (PP2,n)) # x) ^\ n9) <= lim ((P # x) ^\ n9) by A115, A131, RINFSUP2:38; then lim ((ProjMap2 (PP2,n)) # x) <= lim (P # x) by A99, A114; hence ((Partial_Sums F) # x) . n <= lim (P # x) by A138; ::_thesis: verum end; F # x is summable by A1, A5, A23, A96; then A139: Partial_Sums (F # x) is convergent by Def2; (Partial_Sums F) # x is convergent proof percases ( Partial_Sums (F # x) is convergent_to_finite_number or Partial_Sums (F # x) is convergent_to_+infty or Partial_Sums (F # x) is convergent_to_-infty ) by A139, MESFUNC5:def_11; suppose Partial_Sums (F # x) is convergent_to_finite_number ; ::_thesis: (Partial_Sums F) # x is convergent then (Partial_Sums F) # x is convergent_to_finite_number by A2, A3, A23, A96, Th33; hence (Partial_Sums F) # x is convergent by MESFUNC5:def_11; ::_thesis: verum end; suppose Partial_Sums (F # x) is convergent_to_+infty ; ::_thesis: (Partial_Sums F) # x is convergent then (Partial_Sums F) # x is convergent_to_+infty by A2, A3, A23, A96, Th33; hence (Partial_Sums F) # x is convergent by MESFUNC5:def_11; ::_thesis: verum end; suppose Partial_Sums (F # x) is convergent_to_-infty ; ::_thesis: (Partial_Sums F) # x is convergent then (Partial_Sums F) # x is convergent_to_-infty by A2, A3, A23, A96, Th33; hence (Partial_Sums F) # x is convergent by MESFUNC5:def_11; ::_thesis: verum end; end; end; then A140: lim ((Partial_Sums F) # x) <= lim (P # x) by A111, Th9; A141: for k being Element of NAT holds (P # x) . k <= ((Partial_Sums F) # x) . k by A31, A96; (Partial_Sums F) # x is convergent by A3, A4, A23, A96, Th38; then lim (P # x) <= lim ((Partial_Sums F) # x) by A97, A141, RINFSUP2:38; hence lim (P # x) = lim ((Partial_Sums F) # x) by A140, XXREAL_0:1; ::_thesis: verum end; A142: for x being Element of X st x in dom (lim (Partial_Sums F)) holds ( P # x is convergent & lim (P # x) = (lim (Partial_Sums F)) . x ) proof let x be Element of X; ::_thesis: ( x in dom (lim (Partial_Sums F)) implies ( P # x is convergent & lim (P # x) = (lim (Partial_Sums F)) . x ) ) assume A143: x in dom (lim (Partial_Sums F)) ; ::_thesis: ( P # x is convergent & lim (P # x) = (lim (Partial_Sums F)) . x ) then x in dom (lim P) by A1, A6, A22, A37; hence P # x is convergent by A91; ::_thesis: lim (P # x) = (lim (Partial_Sums F)) . x lim (P # x) = lim ((Partial_Sums F) # x) by A1, A23, A37, A93, A143; hence lim (P # x) = (lim (Partial_Sums F)) . x by A143, MESFUNC8:def_9; ::_thesis: verum end; A144: for n being Nat holds P . n is nonnegative proof let n be Nat; ::_thesis: P . n is nonnegative for k being Nat holds (ProjMap2 (FF2,n)) . k is nonnegative by A38; then (Partial_Sums (ProjMap2 (FF2,n))) . n is nonnegative by Th36; hence P . n is nonnegative by A9; ::_thesis: verum end; A145: for x being set st x in dom (lim (Partial_Sums F)) holds (lim (Partial_Sums F)) . x >= 0 proof let x be set ; ::_thesis: ( x in dom (lim (Partial_Sums F)) implies (lim (Partial_Sums F)) . x >= 0 ) assume A146: x in dom (lim (Partial_Sums F)) ; ::_thesis: (lim (Partial_Sums F)) . x >= 0 then reconsider x1 = x as Element of X ; A147: for n being Nat holds ((Partial_Sums F) # x1) . n >= 0 proof let n be Nat; ::_thesis: ((Partial_Sums F) # x1) . n >= 0 (Partial_Sums F) . n is nonnegative by A4, Th36; then ((Partial_Sums F) . n) . x1 >= 0 by SUPINF_2:51; hence ((Partial_Sums F) # x1) . n >= 0 by MESFUNC5:def_13; ::_thesis: verum end; x in dom (F . 0) by A67, A146, Def4; then (Partial_Sums F) # x1 is convergent by A3, A4, Th38; then lim ((Partial_Sums F) # x1) >= 0 by A147, Th10; hence (lim (Partial_Sums F)) . x >= 0 by A146, MESFUNC8:def_9; ::_thesis: verum end; then A148: lim (Partial_Sums F) is nonnegative by SUPINF_2:52; consider I being ExtREAL_sequence such that A149: for n being Nat holds ( I . n = Integral (M,(F . n)) & Integral (M,((Partial_Sums F) . n)) = (Partial_Sums I) . n ) by A1, A2, A3, A4, Th50; for n being set st n in dom I holds 0 <= I . n proof let n be set ; ::_thesis: ( n in dom I implies 0 <= I . n ) assume n in dom I ; ::_thesis: 0 <= I . n then reconsider n1 = n as Nat ; A150: F . n1 is nonnegative by A4; A151: F . n1 is_measurable_on E by A4; E = dom (F . n1) by A1, A3, MESFUNC8:def_2; then 0 <= Integral (M,(F . n1)) by A150, A151, MESFUNC5:90; hence 0 <= I . n by A149; ::_thesis: verum end; then I is V111() by SUPINF_2:52; then A152: Partial_Sums I is non-decreasing by Th16; then A153: Partial_Sums I is convergent by RINFSUP2:37; deffunc H2( Element of NAT ) -> Element of ExtREAL = integral' (M,(P . $1)); consider J being Function of NAT,ExtREAL such that A154: for n being Element of NAT holds J . n = H2(n) from FUNCT_2:sch_4(); reconsider J = J as ExtREAL_sequence ; A155: for n being Nat holds P . n is_simple_func_in S proof let n be Nat; ::_thesis: P . n is_simple_func_in S for m being Nat holds (ProjMap2 (FF2,n)) . m is_simple_func_in S by A10; then (Partial_Sums (ProjMap2 (FF2,n))) . n is_simple_func_in S by Th35; hence P . n is_simple_func_in S by A9; ::_thesis: verum end; A156: for n being Nat holds J . n = integral' (M,(P . n)) proof let n be Nat; ::_thesis: J . n = integral' (M,(P . n)) reconsider n9 = n as Element of NAT by ORDINAL1:def_12; J . n = integral' (M,(P . n9)) by A154; hence J . n = integral' (M,(P . n)) ; ::_thesis: verum end; for n, m being Element of NAT st m <= n holds J . m <= J . n proof let n, m be Element of NAT ; ::_thesis: ( m <= n implies J . m <= J . n ) A157: P . n is nonnegative by A144; A158: P . m is_simple_func_in S by A155; A159: for n, m, l being Element of NAT holds dom ((P . n) - (P . m)) = dom (P . l) proof let n, m, l be Element of NAT ; ::_thesis: dom ((P . n) - (P . m)) = dom (P . l) P . m is_simple_func_in S by A155; then A160: P . m is V120() by MESFUNC5:14; P . n is_simple_func_in S by A155; then P . n is V119() by MESFUNC5:14; then dom ((P . n) - (P . m)) = (dom (P . n)) /\ (dom (P . m)) by A160, MESFUNC5:17; then dom ((P . n) - (P . m)) = (dom (lim (Partial_Sums F))) /\ (dom (P . m)) by A69; then dom ((P . n) - (P . m)) = (dom (lim (Partial_Sums F))) /\ (dom (lim (Partial_Sums F))) by A69; hence dom ((P . n) - (P . m)) = dom (P . l) by A69; ::_thesis: verum end; then A161: dom ((P . n) - (P . m)) = dom (P . n) ; then A162: (P . n) | (dom ((P . n) - (P . m))) = P . n by RELAT_1:68; assume A163: m <= n ; ::_thesis: J . m <= J . n A164: for x being set st x in dom ((P . n) - (P . m)) holds (P . m) . x <= (P . n) . x proof let x be set ; ::_thesis: ( x in dom ((P . n) - (P . m)) implies (P . m) . x <= (P . n) . x ) assume x in dom ((P . n) - (P . m)) ; ::_thesis: (P . m) . x <= (P . n) . x then x in dom (lim (Partial_Sums F)) by A69, A161; hence (P . m) . x <= (P . n) . x by A68, A85, A163; ::_thesis: verum end; A165: P . m is nonnegative by A144; dom ((P . n) - (P . m)) = dom (P . m) by A159; then A166: (P . m) | (dom ((P . n) - (P . m))) = P . m by RELAT_1:68; P . n is_simple_func_in S by A155; then integral' (M,((P . m) | (dom ((P . n) - (P . m))))) <= integral' (M,((P . n) | (dom ((P . n) - (P . m))))) by A157, A158, A165, A164, MESFUNC5:70; then J . m <= integral' (M,(P . n)) by A156, A166, A162; hence J . m <= J . n by A156; ::_thesis: verum end; then J is non-decreasing by RINFSUP2:7; then A167: J is convergent by RINFSUP2:37; A168: for n being Nat holds F . n is V119() by A4, MESFUNC5:12; then A169: for n being Nat holds (Partial_Sums F) . n is_measurable_on E by A4, Th41; then lim (Partial_Sums F) is_measurable_on E by A1, A2, A3, A5, Th44; then integral+ (M,(lim (Partial_Sums F))) = lim J by A68, A155, A85, A156, A69, A142, A144, A148, A167, MESFUNC5:def_15; then A170: Integral (M,(lim (Partial_Sums F))) = lim J by A1, A2, A3, A5, A169, A37, A148, Th44, MESFUNC5:88; A171: for n being Nat for x being Element of X st x in dom (F . n) holds (FF . n) # x is non-decreasing proof let n be Nat; ::_thesis: for x being Element of X st x in dom (F . n) holds (FF . n) # x is non-decreasing let x be Element of X; ::_thesis: ( x in dom (F . n) implies (FF . n) # x is non-decreasing ) assume A172: x in dom (F . n) ; ::_thesis: (FF . n) # x is non-decreasing let i, j be ext-real number ; :: according to VALUED_0:def_15 ::_thesis: ( not i in dom ((FF . n) # x) or not j in dom ((FF . n) # x) or not i <= j or ((FF . n) # x) . i <= ((FF . n) # x) . j ) assume that A173: i in dom ((FF . n) # x) and A174: j in dom ((FF . n) # x) and A175: i <= j ; ::_thesis: ((FF . n) # x) . i <= ((FF . n) # x) . j reconsider i = i, j = j as Element of NAT by A173, A174; ((FF . n) . i) . x <= ((FF . n) . j) . x by A6, A172, A175; then ((FF . n) # x) . i <= ((FF . n) . j) . x by MESFUNC5:def_13; hence ((FF . n) # x) . i <= ((FF . n) # x) . j by MESFUNC5:def_13; ::_thesis: verum end; A176: for n, p being Nat st p >= n holds for x being Element of X st x in E holds ( ((Partial_Sums (ProjMap2 (FF2,p))) . n) . x <= (P . p) . x & (P . p) . x = ((Partial_Sums (ProjMap2 (FF2,p))) . p) . x & ((Partial_Sums (ProjMap2 (FF2,p))) . p) . x <= ((Partial_Sums F) . p) . x & ((Partial_Sums F) . p) . x <= (lim (Partial_Sums F)) . x ) proof let n, p be Nat; ::_thesis: ( p >= n implies for x being Element of X st x in E holds ( ((Partial_Sums (ProjMap2 (FF2,p))) . n) . x <= (P . p) . x & (P . p) . x = ((Partial_Sums (ProjMap2 (FF2,p))) . p) . x & ((Partial_Sums (ProjMap2 (FF2,p))) . p) . x <= ((Partial_Sums F) . p) . x & ((Partial_Sums F) . p) . x <= (lim (Partial_Sums F)) . x ) ) reconsider p1 = p, n1 = n as Element of NAT by ORDINAL1:def_12; assume A177: p >= n ; ::_thesis: for x being Element of X st x in E holds ( ((Partial_Sums (ProjMap2 (FF2,p))) . n) . x <= (P . p) . x & (P . p) . x = ((Partial_Sums (ProjMap2 (FF2,p))) . p) . x & ((Partial_Sums (ProjMap2 (FF2,p))) . p) . x <= ((Partial_Sums F) . p) . x & ((Partial_Sums F) . p) . x <= (lim (Partial_Sums F)) . x ) let x be Element of X; ::_thesis: ( x in E implies ( ((Partial_Sums (ProjMap2 (FF2,p))) . n) . x <= (P . p) . x & (P . p) . x = ((Partial_Sums (ProjMap2 (FF2,p))) . p) . x & ((Partial_Sums (ProjMap2 (FF2,p))) . p) . x <= ((Partial_Sums F) . p) . x & ((Partial_Sums F) . p) . x <= (lim (Partial_Sums F)) . x ) ) A178: for i being Nat holds (ProjMap2 (FF2,p)) . i is nonnegative by A38; assume A179: x in E ; ::_thesis: ( ((Partial_Sums (ProjMap2 (FF2,p))) . n) . x <= (P . p) . x & (P . p) . x = ((Partial_Sums (ProjMap2 (FF2,p))) . p) . x & ((Partial_Sums (ProjMap2 (FF2,p))) . p) . x <= ((Partial_Sums F) . p) . x & ((Partial_Sums F) . p) . x <= (lim (Partial_Sums F)) . x ) then A180: x in dom ((ProjMap2 (FF2,p)) . 0) by A10; ProjMap2 (FF2,p) is with_the_same_dom by A10; then (Partial_Sums (ProjMap2 (FF2,p))) # x is non-decreasing by A180, A178, Th38; then ((Partial_Sums (ProjMap2 (FF2,p))) # x) . n1 <= ((Partial_Sums (ProjMap2 (FF2,p))) # x) . p1 by A177, RINFSUP2:7; then ((Partial_Sums (ProjMap2 (FF2,p))) . n) . x <= ((Partial_Sums (ProjMap2 (FF2,p))) # x) . p by MESFUNC5:def_13; then ((Partial_Sums (ProjMap2 (FF2,p))) . n) . x <= ((Partial_Sums (ProjMap2 (FF2,p))) . p) . x by MESFUNC5:def_13; hence ((Partial_Sums (ProjMap2 (FF2,p))) . n) . x <= (P . p) . x by A9; ::_thesis: ( (P . p) . x = ((Partial_Sums (ProjMap2 (FF2,p))) . p) . x & ((Partial_Sums (ProjMap2 (FF2,p))) . p) . x <= ((Partial_Sums F) . p) . x & ((Partial_Sums F) . p) . x <= (lim (Partial_Sums F)) . x ) thus (P . p) . x = ((Partial_Sums (ProjMap2 (FF2,p))) . p) . x by A9; ::_thesis: ( ((Partial_Sums (ProjMap2 (FF2,p))) . p) . x <= ((Partial_Sums F) . p) . x & ((Partial_Sums F) . p) . x <= (lim (Partial_Sums F)) . x ) A181: ProjMap2 (FF2,p) is additive by A10; A182: ProjMap2 (FF2,p) is with_the_same_dom by A10; A183: for n being Nat for x being Element of X st x in (dom ((ProjMap2 (FF2,p)) . 0)) /\ (dom (F . 0)) holds ((ProjMap2 (FF2,p)) . n) . x <= (F . n) . x proof let n be Nat; ::_thesis: for x being Element of X st x in (dom ((ProjMap2 (FF2,p)) . 0)) /\ (dom (F . 0)) holds ((ProjMap2 (FF2,p)) . n) . x <= (F . n) . x let x be Element of X; ::_thesis: ( x in (dom ((ProjMap2 (FF2,p)) . 0)) /\ (dom (F . 0)) implies ((ProjMap2 (FF2,p)) . n) . x <= (F . n) . x ) reconsider n1 = n as Element of NAT by ORDINAL1:def_12; assume x in (dom ((ProjMap2 (FF2,p)) . 0)) /\ (dom (F . 0)) ; ::_thesis: ((ProjMap2 (FF2,p)) . n) . x <= (F . n) . x then x in dom (F . 0) by XBOOLE_0:def_4; then A184: x in dom (F . n) by A3, MESFUNC8:def_2; then (FF . n) # x is non-decreasing by A171; then lim ((FF . n) # x) = sup ((FF . n) # x) by RINFSUP2:37; then ((FF . n) # x) . p1 <= lim ((FF . n) # x) by RINFSUP2:23; then A185: ((FF . n) # x) . p <= (F . n) . x by A6, A184; ((ProjMap2 (FF2,p)) . n) . x = (FF2 . (n1,p1)) . x by Def9; then ((ProjMap2 (FF2,p)) . n) . x = ((FF . n) . p) . x by A8; hence ((ProjMap2 (FF2,p)) . n) . x <= (F . n) . x by A185, MESFUNC5:def_13; ::_thesis: verum end; x in (dom ((ProjMap2 (FF2,p)) . 0)) /\ (dom (F . 0)) by A1, A179, A180, XBOOLE_0:def_4; hence ((Partial_Sums (ProjMap2 (FF2,p))) . p) . x <= ((Partial_Sums F) . p) . x by A2, A3, A181, A182, A183, Th42; ::_thesis: ((Partial_Sums F) . p) . x <= (lim (Partial_Sums F)) . x (Partial_Sums F) # x is non-decreasing by A1, A3, A4, A179, Th38; then lim ((Partial_Sums F) # x) = sup ((Partial_Sums F) # x) by RINFSUP2:37; then ((Partial_Sums F) # x) . p1 <= lim ((Partial_Sums F) # x) by RINFSUP2:23; then ((Partial_Sums F) . p) . x <= lim ((Partial_Sums F) # x) by MESFUNC5:def_13; hence ((Partial_Sums F) . p) . x <= (lim (Partial_Sums F)) . x by A68, A179, MESFUNC8:def_9; ::_thesis: verum end; for n being Nat holds (Partial_Sums I) . n <= Integral (M,(lim (Partial_Sums F))) proof let n be Nat; ::_thesis: (Partial_Sums I) . n <= Integral (M,(lim (Partial_Sums F))) A186: (Partial_Sums F) . n is nonnegative by A4, Th36; A187: lim (Partial_Sums F) is_measurable_on E by A1, A2, A3, A5, A169, Th44; A188: (Partial_Sums F) . n is_measurable_on E by A4, A168, Th41; A189: E = dom ((Partial_Sums F) . n) by A1, A2, A3, Th29; then for x being Element of X st x in dom ((Partial_Sums F) . n) holds ((Partial_Sums F) . n) . x <= (lim (Partial_Sums F)) . x by A176; then integral+ (M,((Partial_Sums F) . n)) <= integral+ (M,(lim (Partial_Sums F))) by A37, A148, A189, A188, A187, A186, MESFUNC5:85; then Integral (M,((Partial_Sums F) . n)) <= integral+ (M,(lim (Partial_Sums F))) by A169, A189, A186, MESFUNC5:88; then Integral (M,((Partial_Sums F) . n)) <= Integral (M,(lim (Partial_Sums F))) by A37, A145, A187, MESFUNC5:88, SUPINF_2:52; hence (Partial_Sums I) . n <= Integral (M,(lim (Partial_Sums F))) by A149; ::_thesis: verum end; then A190: lim (Partial_Sums I) <= Integral (M,(lim (Partial_Sums F))) by A152, Th9, RINFSUP2:37; take I ; ::_thesis: ( ( for n being Nat holds I . n = Integral (M,((F . n) | E)) ) & I is summable & Integral (M,((lim (Partial_Sums F)) | E)) = Sum I ) thus for n being Nat holds I . n = Integral (M,((F . n) | E)) ::_thesis: ( I is summable & Integral (M,((lim (Partial_Sums F)) | E)) = Sum I ) proof let n be Nat; ::_thesis: I . n = Integral (M,((F . n) | E)) dom (F . 0) = dom (F . n) by A3, MESFUNC8:def_2; then (F . n) | E = F . n by A1, RELAT_1:68; hence I . n = Integral (M,((F . n) | E)) by A149; ::_thesis: verum end; A191: for n being Nat holds J . n = Integral (M,(P . n)) proof let n be Nat; ::_thesis: J . n = Integral (M,(P . n)) A192: P . n is nonnegative by A144; A193: J . n = integral' (M,(P . n)) by A156; P . n is_simple_func_in S by A155; hence J . n = Integral (M,(P . n)) by A192, A193, MESFUNC5:89; ::_thesis: verum end; for n being Element of NAT holds J . n <= (Partial_Sums I) . n proof let n be Element of NAT ; ::_thesis: J . n <= (Partial_Sums I) . n A194: P . n is_measurable_on E by A155, MESFUNC2:34; A195: (Partial_Sums F) . n is nonnegative by A4, Th36; A196: for x being Element of X st x in dom (P . n) holds (P . n) . x <= ((Partial_Sums F) . n) . x proof let x be Element of X; ::_thesis: ( x in dom (P . n) implies (P . n) . x <= ((Partial_Sums F) . n) . x ) assume x in dom (P . n) ; ::_thesis: (P . n) . x <= ((Partial_Sums F) . n) . x then x in dom (lim (Partial_Sums F)) by A69; then (P # x) . n <= ((Partial_Sums F) # x) . n by A1, A23, A37, A31; then (P . n) . x <= ((Partial_Sums F) # x) . n by MESFUNC5:def_13; hence (P . n) . x <= ((Partial_Sums F) . n) . x by MESFUNC5:def_13; ::_thesis: verum end; A197: P . n is nonnegative by A144; A198: dom (P . n) = E by A37, A69; A199: E = dom ((Partial_Sums F) . n) by A1, A2, A3, Th29; (Partial_Sums F) . n is_measurable_on E by A4, A168, Th41; then integral+ (M,(P . n)) <= integral+ (M,((Partial_Sums F) . n)) by A199, A198, A194, A197, A195, A196, MESFUNC5:85; then Integral (M,(P . n)) <= integral+ (M,((Partial_Sums F) . n)) by A144, A198, A194, MESFUNC5:88; then Integral (M,(P . n)) <= Integral (M,((Partial_Sums F) . n)) by A169, A199, A195, MESFUNC5:88; then J . n <= Integral (M,((Partial_Sums F) . n)) by A191; hence J . n <= (Partial_Sums I) . n by A149; ::_thesis: verum end; then lim J <= lim (Partial_Sums I) by A167, A153, RINFSUP2:38; then Sum I = Integral (M,(lim (Partial_Sums F))) by A170, A190, XXREAL_0:1; hence ( I is summable & Integral (M,((lim (Partial_Sums F)) | E)) = Sum I ) by A37, A153, Def2, RELAT_1:68; ::_thesis: verum end; theorem Th51: :: MESFUNC9:51 for X being non empty set for S being SigmaField of X for M being sigma_Measure of S for E being Element of S for F being Functional_Sequence of X,ExtREAL st E c= dom (F . 0) & F is additive & F is with_the_same_dom & ( for n being Nat holds ( F . n is nonnegative & F . n is_measurable_on E ) ) & ( for x being Element of X st x in E holds F # x is summable ) holds ex I being ExtREAL_sequence st ( ( for n being Nat holds I . n = Integral (M,((F . n) | E)) ) & I is summable & Integral (M,((lim (Partial_Sums F)) | E)) = Sum I ) proof let X be non empty set ; ::_thesis: for S being SigmaField of X for M being sigma_Measure of S for E being Element of S for F being Functional_Sequence of X,ExtREAL st E c= dom (F . 0) & F is additive & F is with_the_same_dom & ( for n being Nat holds ( F . n is nonnegative & F . n is_measurable_on E ) ) & ( for x being Element of X st x in E holds F # x is summable ) holds ex I being ExtREAL_sequence st ( ( for n being Nat holds I . n = Integral (M,((F . n) | E)) ) & I is summable & Integral (M,((lim (Partial_Sums F)) | E)) = Sum I ) let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S for E being Element of S for F being Functional_Sequence of X,ExtREAL st E c= dom (F . 0) & F is additive & F is with_the_same_dom & ( for n being Nat holds ( F . n is nonnegative & F . n is_measurable_on E ) ) & ( for x being Element of X st x in E holds F # x is summable ) holds ex I being ExtREAL_sequence st ( ( for n being Nat holds I . n = Integral (M,((F . n) | E)) ) & I is summable & Integral (M,((lim (Partial_Sums F)) | E)) = Sum I ) let M be sigma_Measure of S; ::_thesis: for E being Element of S for F being Functional_Sequence of X,ExtREAL st E c= dom (F . 0) & F is additive & F is with_the_same_dom & ( for n being Nat holds ( F . n is nonnegative & F . n is_measurable_on E ) ) & ( for x being Element of X st x in E holds F # x is summable ) holds ex I being ExtREAL_sequence st ( ( for n being Nat holds I . n = Integral (M,((F . n) | E)) ) & I is summable & Integral (M,((lim (Partial_Sums F)) | E)) = Sum I ) let E be Element of S; ::_thesis: for F being Functional_Sequence of X,ExtREAL st E c= dom (F . 0) & F is additive & F is with_the_same_dom & ( for n being Nat holds ( F . n is nonnegative & F . n is_measurable_on E ) ) & ( for x being Element of X st x in E holds F # x is summable ) holds ex I being ExtREAL_sequence st ( ( for n being Nat holds I . n = Integral (M,((F . n) | E)) ) & I is summable & Integral (M,((lim (Partial_Sums F)) | E)) = Sum I ) let F be Functional_Sequence of X,ExtREAL; ::_thesis: ( E c= dom (F . 0) & F is additive & F is with_the_same_dom & ( for n being Nat holds ( F . n is nonnegative & F . n is_measurable_on E ) ) & ( for x being Element of X st x in E holds F # x is summable ) implies ex I being ExtREAL_sequence st ( ( for n being Nat holds I . n = Integral (M,((F . n) | E)) ) & I is summable & Integral (M,((lim (Partial_Sums F)) | E)) = Sum I ) ) assume that A1: E c= dom (F . 0) and A2: F is additive and A3: F is with_the_same_dom and A4: for n being Nat holds ( F . n is nonnegative & F . n is_measurable_on E ) and A5: for x being Element of X st x in E holds F # x is summable ; ::_thesis: ex I being ExtREAL_sequence st ( ( for n being Nat holds I . n = Integral (M,((F . n) | E)) ) & I is summable & Integral (M,((lim (Partial_Sums F)) | E)) = Sum I ) deffunc H1( Nat) -> Element of bool [:X,ExtREAL:] = (F . $1) | E; consider G being Functional_Sequence of X,ExtREAL such that A6: for n being Nat holds G . n = H1(n) from SEQFUNC:sch_1(); reconsider G = G as with_the_same_dom additive Functional_Sequence of X,ExtREAL by A2, A3, A6, Th18, Th31; A7: for n being Nat holds ( G . n is nonnegative & G . n is_measurable_on E ) proof let n be Nat; ::_thesis: ( G . n is nonnegative & G . n is_measurable_on E ) (F . n) | E is nonnegative by A4, MESFUNC5:15; hence G . n is nonnegative by A6; ::_thesis: G . n is_measurable_on E thus G . n is_measurable_on E by A1, A3, A4, A6, Th20; ::_thesis: verum end; dom ((F . 0) | E) = E by A1, RELAT_1:62; then A8: E = dom (G . 0) by A6; A9: for x being Element of X st x in E holds F # x = G # x proof let x be Element of X; ::_thesis: ( x in E implies F # x = G # x ) assume A10: x in E ; ::_thesis: F # x = G # x for n9 being set st n9 in NAT holds (F # x) . n9 = (G # x) . n9 proof let n9 be set ; ::_thesis: ( n9 in NAT implies (F # x) . n9 = (G # x) . n9 ) assume n9 in NAT ; ::_thesis: (F # x) . n9 = (G # x) . n9 then reconsider n = n9 as Nat ; dom (G . n) = E by A8, MESFUNC8:def_2; then x in dom ((F . n) | E) by A6, A10; then ((F . n) | E) . x = (F . n) . x by FUNCT_1:47; then A11: (G . n) . x = (F . n) . x by A6; (F # x) . n = (F . n) . x by MESFUNC5:def_13; hence (F # x) . n9 = (G # x) . n9 by A11, MESFUNC5:def_13; ::_thesis: verum end; hence F # x = G # x by FUNCT_2:12; ::_thesis: verum end; A12: (lim (Partial_Sums G)) | E = (lim (Partial_Sums F)) | E proof set E1 = dom (F . 0); set PF = Partial_Sums F; set PG = Partial_Sums G; A13: dom (lim (Partial_Sums G)) = dom ((Partial_Sums G) . 0) by MESFUNC8:def_9; dom ((Partial_Sums F) . 0) = dom (F . 0) by A2, A3, Th29; then A14: E c= dom (lim (Partial_Sums F)) by A1, MESFUNC8:def_9; A15: for x being Element of X st x in dom (lim (Partial_Sums G)) holds (lim (Partial_Sums G)) . x = (lim (Partial_Sums F)) . x proof let x be Element of X; ::_thesis: ( x in dom (lim (Partial_Sums G)) implies (lim (Partial_Sums G)) . x = (lim (Partial_Sums F)) . x ) set PFx = Partial_Sums (F # x); set PGx = Partial_Sums (G # x); assume A16: x in dom (lim (Partial_Sums G)) ; ::_thesis: (lim (Partial_Sums G)) . x = (lim (Partial_Sums F)) . x then A17: x in E by A8, A13, Th29; for n being Element of NAT holds ((Partial_Sums G) # x) . n = ((Partial_Sums F) # x) . n proof let n be Element of NAT ; ::_thesis: ((Partial_Sums G) # x) . n = ((Partial_Sums F) # x) . n A18: (Partial_Sums (G # x)) . n = ((Partial_Sums G) # x) . n by A8, A17, Th32; (Partial_Sums (F # x)) . n = ((Partial_Sums F) # x) . n by A1, A2, A3, A17, Th32; hence ((Partial_Sums G) # x) . n = ((Partial_Sums F) # x) . n by A9, A17, A18; ::_thesis: verum end; then A19: lim ((Partial_Sums G) # x) = lim ((Partial_Sums F) # x) by FUNCT_2:63; (lim (Partial_Sums G)) . x = lim ((Partial_Sums G) # x) by A16, MESFUNC8:def_9; hence (lim (Partial_Sums G)) . x = (lim (Partial_Sums F)) . x by A14, A17, A19, MESFUNC8:def_9; ::_thesis: verum end; A20: dom ((Partial_Sums G) . 0) = dom (G . 0) by Th29; then A21: dom ((lim (Partial_Sums G)) | E) = dom (lim (Partial_Sums G)) by A8, A13, RELAT_1:62; A22: dom ((lim (Partial_Sums F)) | E) = E by A14, RELAT_1:62; then A23: dom ((lim (Partial_Sums G)) | E) = dom ((lim (Partial_Sums F)) | E) by A8, A20, A13, RELAT_1:62; for x being Element of X st x in dom ((lim (Partial_Sums G)) | E) holds ((lim (Partial_Sums G)) | E) . x = ((lim (Partial_Sums F)) | E) . x proof let x be Element of X; ::_thesis: ( x in dom ((lim (Partial_Sums G)) | E) implies ((lim (Partial_Sums G)) | E) . x = ((lim (Partial_Sums F)) | E) . x ) assume A24: x in dom ((lim (Partial_Sums G)) | E) ; ::_thesis: ((lim (Partial_Sums G)) | E) . x = ((lim (Partial_Sums F)) | E) . x then A25: ((lim (Partial_Sums F)) | E) . x = (lim (Partial_Sums F)) . x by A23, FUNCT_1:47; (lim (Partial_Sums G)) . x = (lim (Partial_Sums F)) . x by A21, A15, A24; hence ((lim (Partial_Sums G)) | E) . x = ((lim (Partial_Sums F)) | E) . x by A24, A25, FUNCT_1:47; ::_thesis: verum end; hence (lim (Partial_Sums G)) | E = (lim (Partial_Sums F)) | E by A8, A20, A13, A22, PARTFUN1:5, RELAT_1:62; ::_thesis: verum end; for x being Element of X st x in E holds G # x is summable by A1, A5, A6, Th21; then consider I being ExtREAL_sequence such that A26: for n being Nat holds I . n = Integral (M,((G . n) | E)) and A27: I is summable and A28: Integral (M,((lim (Partial_Sums G)) | E)) = Sum I by A8, A7, Lm4; take I ; ::_thesis: ( ( for n being Nat holds I . n = Integral (M,((F . n) | E)) ) & I is summable & Integral (M,((lim (Partial_Sums F)) | E)) = Sum I ) now__::_thesis:_for_n_being_Nat_holds_I_._n_=_Integral_(M,((F_._n)_|_E)) let n be Nat; ::_thesis: I . n = Integral (M,((F . n) | E)) ((F . n) | E) | E = (F . n) | E ; then (G . n) | E = (F . n) | E by A6; hence I . n = Integral (M,((F . n) | E)) by A26; ::_thesis: verum end; hence ( ( for n being Nat holds I . n = Integral (M,((F . n) | E)) ) & I is summable & Integral (M,((lim (Partial_Sums F)) | E)) = Sum I ) by A27, A28, A12; ::_thesis: verum end; theorem :: MESFUNC9:52 for X being non empty set for S being SigmaField of X for M being sigma_Measure of S for E being Element of S for F being Functional_Sequence of X,ExtREAL st E = dom (F . 0) & F . 0 is nonnegative & F is with_the_same_dom & ( for n being Nat holds F . n is_measurable_on E ) & ( for n, m being Nat st n <= m holds for x being Element of X st x in E holds (F . n) . x <= (F . m) . x ) & ( for x being Element of X st x in E holds F # x is convergent ) holds ex I being ExtREAL_sequence st ( ( for n being Nat holds I . n = Integral (M,(F . n)) ) & I is convergent & Integral (M,(lim F)) = lim I ) proof let X be non empty set ; ::_thesis: for S being SigmaField of X for M being sigma_Measure of S for E being Element of S for F being Functional_Sequence of X,ExtREAL st E = dom (F . 0) & F . 0 is nonnegative & F is with_the_same_dom & ( for n being Nat holds F . n is_measurable_on E ) & ( for n, m being Nat st n <= m holds for x being Element of X st x in E holds (F . n) . x <= (F . m) . x ) & ( for x being Element of X st x in E holds F # x is convergent ) holds ex I being ExtREAL_sequence st ( ( for n being Nat holds I . n = Integral (M,(F . n)) ) & I is convergent & Integral (M,(lim F)) = lim I ) let S be SigmaField of X; ::_thesis: for M being sigma_Measure of S for E being Element of S for F being Functional_Sequence of X,ExtREAL st E = dom (F . 0) & F . 0 is nonnegative & F is with_the_same_dom & ( for n being Nat holds F . n is_measurable_on E ) & ( for n, m being Nat st n <= m holds for x being Element of X st x in E holds (F . n) . x <= (F . m) . x ) & ( for x being Element of X st x in E holds F # x is convergent ) holds ex I being ExtREAL_sequence st ( ( for n being Nat holds I . n = Integral (M,(F . n)) ) & I is convergent & Integral (M,(lim F)) = lim I ) let M be sigma_Measure of S; ::_thesis: for E being Element of S for F being Functional_Sequence of X,ExtREAL st E = dom (F . 0) & F . 0 is nonnegative & F is with_the_same_dom & ( for n being Nat holds F . n is_measurable_on E ) & ( for n, m being Nat st n <= m holds for x being Element of X st x in E holds (F . n) . x <= (F . m) . x ) & ( for x being Element of X st x in E holds F # x is convergent ) holds ex I being ExtREAL_sequence st ( ( for n being Nat holds I . n = Integral (M,(F . n)) ) & I is convergent & Integral (M,(lim F)) = lim I ) let E be Element of S; ::_thesis: for F being Functional_Sequence of X,ExtREAL st E = dom (F . 0) & F . 0 is nonnegative & F is with_the_same_dom & ( for n being Nat holds F . n is_measurable_on E ) & ( for n, m being Nat st n <= m holds for x being Element of X st x in E holds (F . n) . x <= (F . m) . x ) & ( for x being Element of X st x in E holds F # x is convergent ) holds ex I being ExtREAL_sequence st ( ( for n being Nat holds I . n = Integral (M,(F . n)) ) & I is convergent & Integral (M,(lim F)) = lim I ) let F be Functional_Sequence of X,ExtREAL; ::_thesis: ( E = dom (F . 0) & F . 0 is nonnegative & F is with_the_same_dom & ( for n being Nat holds F . n is_measurable_on E ) & ( for n, m being Nat st n <= m holds for x being Element of X st x in E holds (F . n) . x <= (F . m) . x ) & ( for x being Element of X st x in E holds F # x is convergent ) implies ex I being ExtREAL_sequence st ( ( for n being Nat holds I . n = Integral (M,(F . n)) ) & I is convergent & Integral (M,(lim F)) = lim I ) ) assume that A1: E = dom (F . 0) and A2: F . 0 is nonnegative and A3: F is with_the_same_dom and A4: for n being Nat holds F . n is_measurable_on E and A5: for n, m being Nat st n <= m holds for x being Element of X st x in E holds (F . n) . x <= (F . m) . x and A6: for x being Element of X st x in E holds F # x is convergent ; ::_thesis: ex I being ExtREAL_sequence st ( ( for n being Nat holds I . n = Integral (M,(F . n)) ) & I is convergent & Integral (M,(lim F)) = lim I ) A7: lim F is_measurable_on E by A1, A3, A4, A6, MESFUNC8:25; A8: for n being Nat holds F . n is nonnegative proof let n be Nat; ::_thesis: F . n is nonnegative for x being set st x in dom (F . n) holds 0 <= (F . n) . x proof let x be set ; ::_thesis: ( x in dom (F . n) implies 0 <= (F . n) . x ) assume x in dom (F . n) ; ::_thesis: 0 <= (F . n) . x then x in E by A1, A3, MESFUNC8:def_2; then (F . 0) . x <= (F . n) . x by A5; hence 0 <= (F . n) . x by A2, SUPINF_2:51; ::_thesis: verum end; hence F . n is nonnegative by SUPINF_2:52; ::_thesis: verum end; percases ( ex n being Nat st M . (E /\ (eq_dom ((F . n),+infty))) <> 0 or for n being Nat holds M . (E /\ (eq_dom ((F . n),+infty))) = 0 ) ; suppose ex n being Nat st M . (E /\ (eq_dom ((F . n),+infty))) <> 0 ; ::_thesis: ex I being ExtREAL_sequence st ( ( for n being Nat holds I . n = Integral (M,(F . n)) ) & I is convergent & Integral (M,(lim F)) = lim I ) then consider N being Nat such that A9: M . (E /\ (eq_dom ((F . N),+infty))) <> 0 ; A10: E = dom (F . N) by A1, A3, MESFUNC8:def_2; then reconsider EE = E /\ (eq_dom ((F . N),+infty)) as Element of S by A4, MESFUNC1:33; A11: EE c= E by XBOOLE_1:17; then A12: F . N is_measurable_on EE by A4, MESFUNC1:30; EE c= dom (F . N) by A1, A3, A11, MESFUNC8:def_2; then A13: EE = dom ((F . N) | EE) by RELAT_1:62; then EE = (dom (F . N)) /\ EE by RELAT_1:61; then A14: (F . N) | EE is_measurable_on EE by A12, MESFUNC5:42; now__::_thesis:_for_x_being_set_st_x_in_EE_holds_ x_in_eq_dom_(((F_._N)_|_EE),+infty) let x be set ; ::_thesis: ( x in EE implies x in eq_dom (((F . N) | EE),+infty) ) assume A15: x in EE ; ::_thesis: x in eq_dom (((F . N) | EE),+infty) then x in eq_dom ((F . N),+infty) by XBOOLE_0:def_4; then (F . N) . x = +infty by MESFUNC1:def_15; then ((F . N) | EE) . x = +infty by A13, A15, FUNCT_1:47; hence x in eq_dom (((F . N) | EE),+infty) by A13, A15, MESFUNC1:def_15; ::_thesis: verum end; then A16: EE c= eq_dom (((F . N) | EE),+infty) by TARSKI:def_3; for x being set st x in eq_dom (((F . N) | EE),+infty) holds x in EE by A13, MESFUNC1:def_15; then eq_dom (((F . N) | EE),+infty) c= EE by TARSKI:def_3; then EE = eq_dom (((F . N) | EE),+infty) by A16, XBOOLE_0:def_10; then A17: M . (EE /\ (eq_dom (((F . N) | EE),+infty))) <> 0 by A9; F . N is_measurable_on E by A4; then A18: Integral (M,((F . N) | EE)) <= Integral (M,((F . N) | E)) by A8, A10, A11, MESFUNC5:93; reconsider N1 = N as Element of NAT by ORDINAL1:def_12; deffunc H1( Element of NAT ) -> Element of ExtREAL = Integral (M,(F . $1)); consider I being Function of NAT,ExtREAL such that A19: for n being Element of NAT holds I . n = H1(n) from FUNCT_2:sch_4(); reconsider I = I as ExtREAL_sequence ; A20: 0 < M . (E /\ (eq_dom ((F . N),+infty))) by A9, SUPINF_2:51; A21: for n being Nat holds I . n = Integral (M,(F . n)) proof let n be Nat; ::_thesis: I . n = Integral (M,(F . n)) reconsider n1 = n as Element of NAT by ORDINAL1:def_12; I . n = Integral (M,(F . n1)) by A19; hence I . n = Integral (M,(F . n)) ; ::_thesis: verum end; take I ; ::_thesis: ( ( for n being Nat holds I . n = Integral (M,(F . n)) ) & I is convergent & Integral (M,(lim F)) = lim I ) A22: dom (lim F) = dom (F . 0) by MESFUNC8:def_9; for x being set st x in dom (lim F) holds (lim F) . x >= 0 proof let x be set ; ::_thesis: ( x in dom (lim F) implies (lim F) . x >= 0 ) assume A23: x in dom (lim F) ; ::_thesis: (lim F) . x >= 0 then reconsider x1 = x as Element of X ; for n being Nat holds (F # x1) . n >= 0 proof let n be Nat; ::_thesis: (F # x1) . n >= 0 A24: (F . 0) . x1 >= 0 by A2, SUPINF_2:51; (F . n) . x1 >= (F . 0) . x1 by A1, A5, A22, A23; hence (F # x1) . n >= 0 by A24, MESFUNC5:def_13; ::_thesis: verum end; then lim (F # x1) >= 0 by A1, A6, A22, A23, Th10; hence (lim F) . x >= 0 by A23, MESFUNC8:def_9; ::_thesis: verum end; then A25: lim F is nonnegative by SUPINF_2:52; A26: E = dom (lim F) by A1, MESFUNC8:def_9; A27: EE c= E /\ (eq_dom ((lim F),+infty)) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in EE or x in E /\ (eq_dom ((lim F),+infty)) ) assume A28: x in EE ; ::_thesis: x in E /\ (eq_dom ((lim F),+infty)) then reconsider x1 = x as Element of X ; x in eq_dom ((F . N),+infty) by A28, XBOOLE_0:def_4; then (F . N) . x1 = +infty by MESFUNC1:def_15; then A29: (F # x1) . N = +infty by MESFUNC5:def_13; A30: x in E by A28, XBOOLE_0:def_4; for n, m being Element of NAT st m <= n holds (F # x1) . m <= (F # x1) . n proof let n, m be Element of NAT ; ::_thesis: ( m <= n implies (F # x1) . m <= (F # x1) . n ) assume m <= n ; ::_thesis: (F # x1) . m <= (F # x1) . n then (F . m) . x1 <= (F . n) . x1 by A5, A30; then (F # x1) . m <= (F . n) . x1 by MESFUNC5:def_13; hence (F # x1) . m <= (F # x1) . n by MESFUNC5:def_13; ::_thesis: verum end; then A31: F # x1 is non-decreasing by RINFSUP2:7; then A32: (F # x1) ^\ N1 is non-decreasing by RINFSUP2:26; ((F # x1) ^\ N1) . 0 = (F # x1) . (0 + N) by NAT_1:def_3; then for n being Element of NAT holds +infty <= ((F # x1) ^\ N1) . n by A29, A32, RINFSUP2:7; then (F # x1) ^\ N1 is convergent_to_+infty by RINFSUP2:32; then A33: lim ((F # x1) ^\ N1) = +infty by Th7; A34: sup (F # x1) = sup ((F # x1) ^\ N1) by A31, RINFSUP2:26; lim (F # x1) = sup (F # x1) by A31, RINFSUP2:37; then lim (F # x1) = +infty by A32, A34, A33, RINFSUP2:37; then (lim F) . x1 = +infty by A1, A22, A30, MESFUNC8:def_9; then x in eq_dom ((lim F),+infty) by A26, A30, MESFUNC1:def_15; hence x in E /\ (eq_dom ((lim F),+infty)) by A30, XBOOLE_0:def_4; ::_thesis: verum end; A35: for n, m being Element of NAT st m <= n holds I . m <= I . n proof let n, m be Element of NAT ; ::_thesis: ( m <= n implies I . m <= I . n ) A36: F . m is_measurable_on E by A4; assume m <= n ; ::_thesis: I . m <= I . n then A37: for x being Element of X st x in E holds (F . m) . x <= (F . n) . x by A5; A38: E = dom (F . m) by A1, A3, MESFUNC8:def_2; A39: E = dom (F . n) by A1, A3, MESFUNC8:def_2; F . n is_measurable_on E by A4; then Integral (M,((F . m) | E)) <= Integral (M,((F . n) | E)) by A8, A38, A39, A36, A37, Th15; then Integral (M,(F . m)) <= Integral (M,((F . n) | E)) by A38, RELAT_1:68; then Integral (M,(F . m)) <= Integral (M,(F . n)) by A39, RELAT_1:68; then I . m <= Integral (M,(F . n)) by A19; hence I . m <= I . n by A19; ::_thesis: verum end; then A40: I is non-decreasing by RINFSUP2:7; then A41: I ^\ N1 is non-decreasing by RINFSUP2:26; F . N is nonnegative by A8; then Integral (M,((F . N) | EE)) = +infty by A13, A14, A17, Th13, MESFUNC5:15; then +infty <= Integral (M,(F . N)) by A10, A18, RELAT_1:69; then A42: Integral (M,(F . N)) = +infty by XXREAL_0:4; for k being Element of NAT holds +infty <= (I ^\ N1) . k proof let k be Element of NAT ; ::_thesis: +infty <= (I ^\ N1) . k I . N1 <= I . (N1 + k) by A35, NAT_1:12; then I . N1 <= (I ^\ N1) . k by NAT_1:def_3; hence +infty <= (I ^\ N1) . k by A42, A21; ::_thesis: verum end; then I ^\ N1 is convergent_to_+infty by RINFSUP2:32; then A43: lim (I ^\ N1) = +infty by Th7; E /\ (eq_dom ((lim F),+infty)) is Element of S by A7, A26, MESFUNC1:33; then A44: M . (E /\ (eq_dom ((lim F),+infty))) <> 0 by A27, A20, MEASURE1:31; A45: sup I = sup (I ^\ N1) by A40, RINFSUP2:26; lim I = sup I by A40, RINFSUP2:37; then lim I = +infty by A41, A45, A43, RINFSUP2:37; hence ( ( for n being Nat holds I . n = Integral (M,(F . n)) ) & I is convergent & Integral (M,(lim F)) = lim I ) by A7, A21, A40, A26, A25, A44, Th13, RINFSUP2:37; ::_thesis: verum end; supposeA46: for n being Nat holds M . (E /\ (eq_dom ((F . n),+infty))) = 0 ; ::_thesis: ex I being ExtREAL_sequence st ( ( for n being Nat holds I . n = Integral (M,(F . n)) ) & I is convergent & Integral (M,(lim F)) = lim I ) defpred S1[ Element of NAT , set ] means $2 = E /\ (eq_dom ((F . $1),+infty)); A47: for n being Element of NAT ex A being Element of S st S1[n,A] proof let n be Element of NAT ; ::_thesis: ex A being Element of S st S1[n,A] E c= dom (F . n) by A1, A3, MESFUNC8:def_2; then reconsider A = E /\ (eq_dom ((F . n),+infty)) as Element of S by A4, MESFUNC1:33; take A ; ::_thesis: S1[n,A] thus S1[n,A] ; ::_thesis: verum end; consider L being Function of NAT,S such that A48: for n being Element of NAT holds S1[n,L . n] from FUNCT_2:sch_3(A47); A49: rng L c= S by RELAT_1:def_19; rng L is N_Sub_set_fam of X by MEASURE1:23; then reconsider E0 = rng L as N_Measure_fam of S by A49, MEASURE2:def_1; set E1 = E \ (union E0); deffunc H1( Nat) -> Element of bool [:X,ExtREAL:] = (F . $1) | (E \ (union E0)); consider H being Functional_Sequence of X,ExtREAL such that A50: for n being Nat holds H . n = H1(n) from SEQFUNC:sch_1(); deffunc H2( Element of NAT ) -> Element of ExtREAL = Integral (M,((F . $1) | (E \ (union E0)))); consider I being Function of NAT,ExtREAL such that A51: for n being Element of NAT holds I . n = H2(n) from FUNCT_2:sch_4(); reconsider I = I as ExtREAL_sequence ; A52: E \ (union E0) c= E by XBOOLE_1:36; then A53: for n being Nat holds F . n is_measurable_on E \ (union E0) by A4, MESFUNC1:30; A54: for n being Nat holds ( dom (H . n) = E \ (union E0) & H . n is V119() & H . n is V120() ) proof let n be Nat; ::_thesis: ( dom (H . n) = E \ (union E0) & H . n is V119() & H . n is V120() ) A55: dom (H . n) = dom ((F . n) | (E \ (union E0))) by A50; E \ (union E0) c= dom (F . n) by A1, A3, A52, MESFUNC8:def_2; hence dom (H . n) = E \ (union E0) by A55, RELAT_1:62; ::_thesis: ( H . n is V119() & H . n is V120() ) (F . n) | (E \ (union E0)) is nonnegative by A8, MESFUNC5:15; then H . n is nonnegative by A50; hence H . n is V119() by MESFUNC5:12; ::_thesis: H . n is V120() for x being set st x in dom (H . n) holds (H . n) . x < +infty proof reconsider n1 = n as Element of NAT by ORDINAL1:def_12; let x be set ; ::_thesis: ( x in dom (H . n) implies (H . n) . x < +infty ) A56: L . n = E /\ (eq_dom ((F . n1),+infty)) by A48; dom L = NAT by FUNCT_2:def_1; then A57: L . n in rng L by A56, FUNCT_1:3; assume x in dom (H . n) ; ::_thesis: (H . n) . x < +infty then A58: x in dom ((F . n) | (E \ (union E0))) by A50; then A59: x in E \ (union E0) by RELAT_1:57; A60: x in dom (F . n) by A58, RELAT_1:57; assume A61: (H . n) . x >= +infty ; ::_thesis: contradiction (H . n) . x = ((F . n) | (E \ (union E0))) . x by A50; then (H . n) . x = (F . n) . x by A59, FUNCT_1:49; then (F . n) . x = +infty by A61, XXREAL_0:4; then x in eq_dom ((F . n),+infty) by A60, MESFUNC1:def_15; then x in L . n by A52, A59, A56, XBOOLE_0:def_4; then x in union E0 by A57, TARSKI:def_4; hence contradiction by A59, XBOOLE_0:def_5; ::_thesis: verum end; hence H . n is V120() by MESFUNC5:11; ::_thesis: verum end; for n, m being Nat holds dom (H . n) = dom (H . m) proof let n, m be Nat; ::_thesis: dom (H . n) = dom (H . m) dom (H . n) = E \ (union E0) by A54; hence dom (H . n) = dom (H . m) by A54; ::_thesis: verum end; then reconsider H = H as with_the_same_dom Functional_Sequence of X,ExtREAL by MESFUNC8:def_2; defpred S2[ Element of NAT , set , set ] means $3 = (H . ($1 + 1)) - (H . $1); A62: for n being Element of NAT for x being set ex y being set st S2[n,x,y] ; consider G being Function such that A63: ( dom G = NAT & G . 0 = H . 0 & ( for n being Element of NAT holds S2[n,G . n,G . (n + 1)] ) ) from RECDEF_1:sch_1(A62); A64: for n being Nat holds G . (n + 1) = (H . (n + 1)) - (H . n) proof let n be Nat; ::_thesis: G . (n + 1) = (H . (n + 1)) - (H . n) reconsider n1 = n as Element of NAT by ORDINAL1:def_12; G . (n + 1) = (H . (n1 + 1)) - (H . n) by A63; hence G . (n + 1) = (H . (n + 1)) - (H . n) ; ::_thesis: verum end; now__::_thesis:_for_f_being_set_st_f_in_rng_G_holds_ f_in_PFuncs_(X,ExtREAL) defpred S3[ Nat] means G . $1 is PartFunc of X,ExtREAL; let f be set ; ::_thesis: ( f in rng G implies f in PFuncs (X,ExtREAL) ) assume f in rng G ; ::_thesis: f in PFuncs (X,ExtREAL) then consider m being set such that A65: m in dom G and A66: f = G . m by FUNCT_1:def_3; reconsider m = m as Nat by A63, A65; A67: for n being Nat st S3[n] holds S3[n + 1] proof let n be Nat; ::_thesis: ( S3[n] implies S3[n + 1] ) assume S3[n] ; ::_thesis: S3[n + 1] G . (n + 1) = (H . (n + 1)) - (H . n) by A64; hence S3[n + 1] ; ::_thesis: verum end; A68: S3[ 0 ] by A63; for n being Nat holds S3[n] from NAT_1:sch_2(A68, A67); then G . m is PartFunc of X,ExtREAL ; hence f in PFuncs (X,ExtREAL) by A66, PARTFUN1:45; ::_thesis: verum end; then rng G c= PFuncs (X,ExtREAL) by TARSKI:def_3; then reconsider G = G as Functional_Sequence of X,ExtREAL by A63, FUNCT_2:def_1, RELSET_1:4; A69: for n being Nat holds dom (G . n) = E \ (union E0) proof let n be Nat; ::_thesis: dom (G . n) = E \ (union E0) now__::_thesis:_(_n_<>_0_implies_dom_(G_._n)_=_E_\_(union_E0)_) assume n <> 0 ; ::_thesis: dom (G . n) = E \ (union E0) then consider k being Nat such that A70: n = k + 1 by NAT_1:6; A71: H . (k + 1) is V119() by A54; A72: H . k is V120() by A54; G . (k + 1) = (H . (k + 1)) - (H . k) by A64; then dom (G . (k + 1)) = (dom (H . (k + 1))) /\ (dom (H . k)) by A71, A72, MESFUNC5:17; then dom (G . (k + 1)) = (E \ (union E0)) /\ (dom (H . k)) by A54; then dom (G . (k + 1)) = (E \ (union E0)) /\ (E \ (union E0)) by A54; hence dom (G . n) = E \ (union E0) by A70; ::_thesis: verum end; hence dom (G . n) = E \ (union E0) by A54, A63; ::_thesis: verum end; A73: for n, m being Nat holds dom (G . n) = dom (G . m) proof let n, m be Nat; ::_thesis: dom (G . n) = dom (G . m) dom (G . n) = E \ (union E0) by A69; hence dom (G . n) = dom (G . m) by A69; ::_thesis: verum end; A74: for n being Nat holds G . n is nonnegative proof let n be Nat; ::_thesis: G . n is nonnegative A75: ( n <> 0 implies G . n is nonnegative ) proof assume n <> 0 ; ::_thesis: G . n is nonnegative then consider k being Nat such that A76: n = k + 1 by NAT_1:6; A77: G . (k + 1) = (H . (k + 1)) - (H . k) by A64; for x being set st x in dom (G . (k + 1)) holds 0 <= (G . (k + 1)) . x proof let x be set ; ::_thesis: ( x in dom (G . (k + 1)) implies 0 <= (G . (k + 1)) . x ) assume A78: x in dom (G . (k + 1)) ; ::_thesis: 0 <= (G . (k + 1)) . x A79: dom (G . (k + 1)) = E \ (union E0) by A69; (H . k) . x = ((F . k) | (E \ (union E0))) . x by A50; then A80: (H . k) . x = (F . k) . x by A78, A79, FUNCT_1:49; (H . (k + 1)) . x = ((F . (k + 1)) | (E \ (union E0))) . x by A50; then A81: (H . (k + 1)) . x = (F . (k + 1)) . x by A78, A79, FUNCT_1:49; (F . k) . x <= (F . (k + 1)) . x by A5, A52, A78, A79, NAT_1:11; then ((H . (k + 1)) . x) - ((H . k) . x) >= 0 by A81, A80, XXREAL_3:40; hence 0 <= (G . (k + 1)) . x by A77, A78, MESFUNC1:def_4; ::_thesis: verum end; hence G . n is nonnegative by A76, SUPINF_2:52; ::_thesis: verum end; ( n = 0 implies G . n is nonnegative ) proof assume A82: n = 0 ; ::_thesis: G . n is nonnegative (F . n) | (E \ (union E0)) is nonnegative by A8, MESFUNC5:15; hence G . n is nonnegative by A50, A63, A82; ::_thesis: verum end; hence G . n is nonnegative by A75; ::_thesis: verum end; A83: for n1 being set st n1 in NAT holds H . n1 = (Partial_Sums G) . n1 proof defpred S3[ Nat] means H . $1 = (Partial_Sums G) . $1; let n1 be set ; ::_thesis: ( n1 in NAT implies H . n1 = (Partial_Sums G) . n1 ) assume n1 in NAT ; ::_thesis: H . n1 = (Partial_Sums G) . n1 then reconsider n = n1 as Nat ; A84: for k being Nat st S3[k] holds S3[k + 1] proof let k be Nat; ::_thesis: ( S3[k] implies S3[k + 1] ) A85: H . k is V120() by A54; A86: H . k is V119() by A54; A87: dom (G . (k + 1)) = E \ (union E0) by A69; G . (k + 1) is V119() by A74, MESFUNC5:12; then dom ((G . (k + 1)) + (H . k)) = (dom (G . (k + 1))) /\ (dom (H . k)) by A86, MESFUNC5:16; then dom ((G . (k + 1)) + (H . k)) = (E \ (union E0)) /\ (E \ (union E0)) by A54, A87; then A88: dom (H . (k + 1)) = dom ((G . (k + 1)) + (H . k)) by A54; A89: G . (k + 1) = (H . (k + 1)) - (H . k) by A64; for x being Element of X st x in dom (H . (k + 1)) holds (H . (k + 1)) . x = ((G . (k + 1)) + (H . k)) . x proof let x be Element of X; ::_thesis: ( x in dom (H . (k + 1)) implies (H . (k + 1)) . x = ((G . (k + 1)) + (H . k)) . x ) A90: (H . k) . x <> +infty by A85, MESFUNC5:def_6; (H . k) . x <> -infty by A86, MESFUNC5:def_5; then (((H . (k + 1)) . x) - ((H . k) . x)) + ((H . k) . x) = ((H . (k + 1)) . x) - (((H . k) . x) - ((H . k) . x)) by A90, XXREAL_3:32; then (((H . (k + 1)) . x) - ((H . k) . x)) + ((H . k) . x) = ((H . (k + 1)) . x) - 0. by XXREAL_3:7; then A91: (((H . (k + 1)) . x) - ((H . k) . x)) + ((H . k) . x) = (H . (k + 1)) . x by XXREAL_3:4; assume A92: x in dom (H . (k + 1)) ; ::_thesis: (H . (k + 1)) . x = ((G . (k + 1)) + (H . k)) . x then x in E \ (union E0) by A54; then (H . (k + 1)) . x = ((G . (k + 1)) . x) + ((H . k) . x) by A89, A87, A91, MESFUNC1:def_4; hence (H . (k + 1)) . x = ((G . (k + 1)) + (H . k)) . x by A88, A92, MESFUNC1:def_3; ::_thesis: verum end; then A93: H . (k + 1) = (G . (k + 1)) + (H . k) by A88, PARTFUN1:5; assume S3[k] ; ::_thesis: S3[k + 1] hence S3[k + 1] by A93, Def4; ::_thesis: verum end; A94: S3[ 0 ] by A63, Def4; for k being Nat holds S3[k] from NAT_1:sch_2(A94, A84); then H . n = (Partial_Sums G) . n ; hence H . n1 = (Partial_Sums G) . n1 ; ::_thesis: verum end; then A95: for n being Nat holds ( H . n = (Partial_Sums G) . n & lim H = lim (Partial_Sums G) ) by FUNCT_2:12; reconsider G = G as with_the_same_dom Functional_Sequence of X,ExtREAL by A73, MESFUNC8:def_2; reconsider G = G as with_the_same_dom additive Functional_Sequence of X,ExtREAL by A74, Th30; A96: for k being Nat holds H . k is V63() proof let k be Nat; ::_thesis: H . k is V63() for x being Element of X st x in dom (H . k) holds |.((H . k) . x).| < +infty proof let x be Element of X; ::_thesis: ( x in dom (H . k) implies |.((H . k) . x).| < +infty ) assume x in dom (H . k) ; ::_thesis: |.((H . k) . x).| < +infty H . k is V120() by A54; then A97: (H . k) . x < +infty by MESFUNC5:def_6; H . k is V119() by A54; then -infty < (H . k) . x by MESFUNC5:def_5; hence |.((H . k) . x).| < +infty by A97, EXTREAL2:29, XXREAL_0:4; ::_thesis: verum end; hence H . k is V63() by MESFUNC2:def_1; ::_thesis: verum end; A98: for n being Nat holds G . n is_measurable_on E \ (union E0) proof let n be Nat; ::_thesis: G . n is_measurable_on E \ (union E0) ( n <> 0 implies G . n is_measurable_on E \ (union E0) ) proof assume n <> 0 ; ::_thesis: G . n is_measurable_on E \ (union E0) then consider k being Nat such that A99: n = k + 1 by NAT_1:6; A100: E \ (union E0) = dom (H . k) by A54; A101: G . (k + 1) = (H . (k + 1)) - (H . k) by A64; A102: H . k is V63() by A96; A103: H . k is_measurable_on E \ (union E0) by A1, A3, A53, A50, Th20, XBOOLE_1:36; A104: H . (k + 1) is V63() by A96; H . (k + 1) is_measurable_on E \ (union E0) by A1, A3, A53, A50, Th20, XBOOLE_1:36; hence G . n is_measurable_on E \ (union E0) by A99, A103, A100, A104, A102, A101, MESFUNC2:11; ::_thesis: verum end; hence G . n is_measurable_on E \ (union E0) by A1, A3, A52, A53, A50, A63, Th20; ::_thesis: verum end; A105: E \ (union E0) = dom (G . 0) by A54, A63; for x being Element of X st x in E \ (union E0) holds G # x is summable proof let x be Element of X; ::_thesis: ( x in E \ (union E0) implies G # x is summable ) assume A106: x in E \ (union E0) ; ::_thesis: G # x is summable E \ (union E0) c= E by XBOOLE_1:36; then F # x is convergent by A6, A106; then H # x is convergent by A50, A106, Th12; then (Partial_Sums G) # x is convergent by A83, FUNCT_2:12; then Partial_Sums (G # x) is convergent by A105, A106, Th33; hence G # x is summable by Def2; ::_thesis: verum end; then consider J being ExtREAL_sequence such that A107: for n being Nat holds J . n = Integral (M,((G . n) | (E \ (union E0)))) and J is summable and A108: Integral (M,((lim (Partial_Sums G)) | (E \ (union E0)))) = Sum J by A74, A105, A98, Th51; for n being set st n in NAT holds I . n = (Partial_Sums J) . n proof let n be set ; ::_thesis: ( n in NAT implies I . n = (Partial_Sums J) . n ) assume n in NAT ; ::_thesis: I . n = (Partial_Sums J) . n then reconsider n1 = n as Element of NAT ; A109: for n being Nat holds J . n = Integral (M,(G . n)) proof let n be Nat; ::_thesis: J . n = Integral (M,(G . n)) dom (G . n) = E \ (union E0) by A69; then (G . n) | (E \ (union E0)) = G . n by RELAT_1:68; hence J . n = Integral (M,(G . n)) by A107; ::_thesis: verum end; E \ (union E0) = dom (G . 0) by A69; then (Partial_Sums J) . n1 = Integral (M,((Partial_Sums G) . n1)) by A74, A98, A109, Th46; then (Partial_Sums J) . n1 = Integral (M,(H . n1)) by A83; then (Partial_Sums J) . n1 = Integral (M,((F . n1) | (E \ (union E0)))) by A50; hence I . n = (Partial_Sums J) . n by A51; ::_thesis: verum end; then A110: I = Partial_Sums J by FUNCT_2:12; dom (lim (Partial_Sums G)) = dom (H . 0) by A95, MESFUNC8:def_9; then dom (lim (Partial_Sums G)) = E \ (union E0) by A54; then A111: lim I = Integral (M,(lim H)) by A95, A108, A110, RELAT_1:68; take I ; ::_thesis: ( ( for n being Nat holds I . n = Integral (M,(F . n)) ) & I is convergent & Integral (M,(lim F)) = lim I ) A112: for x being Element of X st x in E \ (union E0) holds F # x is convergent proof let x be Element of X; ::_thesis: ( x in E \ (union E0) implies F # x is convergent ) A113: E \ (union E0) c= E by XBOOLE_1:36; assume x in E \ (union E0) ; ::_thesis: F # x is convergent hence F # x is convergent by A6, A113; ::_thesis: verum end; A114: for n being Element of NAT st 0 <= n holds (M * L) . n = 0 proof let n be Element of NAT ; ::_thesis: ( 0 <= n implies (M * L) . n = 0 ) assume 0 <= n ; ::_thesis: (M * L) . n = 0 dom L = NAT by FUNCT_2:def_1; then (M * L) . n = M . (L . n) by FUNCT_1:13; then (M * L) . n = M . (E /\ (eq_dom ((F . n),+infty))) by A48; hence (M * L) . n = 0 by A46; ::_thesis: verum end; M * L is V111() by MEASURE2:1; then SUM (M * L) = (Ser (M * L)) . 0 by A114, SUPINF_2:48; then SUM (M * L) = (M * L) . 0 by SUPINF_2:44; then SUM (M * L) = 0 by A114; then M . (union E0) <= 0 by MEASURE2:11; then A115: M . (union E0) = 0 by SUPINF_2:51; A116: for n being Nat holds I . n = Integral (M,(F . n)) proof let n be Nat; ::_thesis: I . n = Integral (M,(F . n)) reconsider n1 = n as Element of NAT by ORDINAL1:def_12; A117: I . n = Integral (M,((F . n1) | (E \ (union E0)))) by A51; dom (F . n) = E by A1, A3, MESFUNC8:def_2; hence I . n = Integral (M,(F . n)) by A4, A115, A117, MESFUNC5:95; ::_thesis: verum end; for n, m being Element of NAT st m <= n holds I . m <= I . n proof let n, m be Element of NAT ; ::_thesis: ( m <= n implies I . m <= I . n ) A118: F . m is nonnegative by A8; A119: dom (F . m) = E by A1, A3, MESFUNC8:def_2; assume m <= n ; ::_thesis: I . m <= I . n then A120: for x being Element of X st x in dom (F . m) holds (F . m) . x <= (F . n) . x by A5, A119; A121: dom (F . n) = E by A1, A3, MESFUNC8:def_2; A122: F . n is_measurable_on E by A4; A123: F . n is nonnegative by A8; F . m is_measurable_on E by A4; then integral+ (M,(F . m)) <= integral+ (M,(F . n)) by A119, A121, A120, A118, A123, A122, MESFUNC5:85; then Integral (M,(F . m)) <= integral+ (M,(F . n)) by A4, A119, A118, MESFUNC5:88; then Integral (M,(F . m)) <= Integral (M,(F . n)) by A4, A121, A123, MESFUNC5:88; then I . m <= Integral (M,(F . n)) by A116; hence I . m <= I . n by A116; ::_thesis: verum end; then A124: I is non-decreasing by RINFSUP2:7; E = dom (lim F) by A1, MESFUNC8:def_9; then A125: Integral (M,(lim F)) = Integral (M,((lim F) | (E \ (union E0)))) by A7, A115, MESFUNC5:95; E \ (union E0) c= dom (F . 0) by A1, XBOOLE_1:36; hence ( ( for n being Nat holds I . n = Integral (M,(F . n)) ) & I is convergent & Integral (M,(lim F)) = lim I ) by A125, A50, A116, A124, A112, A111, Th19, RINFSUP2:37; ::_thesis: verum end; end; end;