:: 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;