
theorem Th105:
  for X be non empty set, S be SigmaField of X, M be
sigma_Measure of S, f be PartFunc of X,ExtREAL st f is_integrable_on M holds f"
  {+infty} in S & f"{-infty} in S & M.(f"{+infty})=0 & M.(f"{-infty})=0 & f"{
  +infty} \/ f"{-infty} in S & M.(f"{+infty} \/ f"{-infty})=0
proof
  let X be non empty set;
  let S be SigmaField of X;
  let M be sigma_Measure of S;
  let f be PartFunc of X,ExtREAL;
A1: max+f is nonnegative by Lm1;
  assume
A2: f is_integrable_on M;
  then
A3: integral+(M,max+f) < +infty;
  consider A be Element of S such that
A4: A = dom f and
A5: f is A-measurable by A2;
A6: for x be object holds ( x in eq_dom(f,+infty) implies x in A ) & ( x in
  eq_dom(f,-infty) implies x in A ) by A4,MESFUNC1:def 15;
  then
A7: eq_dom(f,+infty) c= A;
  then
A8: A /\ eq_dom(f,+infty) = eq_dom(f,+infty) by XBOOLE_1:28;
A9: eq_dom(f,-infty) c= A by A6;
  then
A10: A /\ eq_dom(f,-infty) = eq_dom(f,-infty) by XBOOLE_1:28;
A11: A /\ eq_dom(f,+infty) in S by A4,A5,MESFUNC1:33;
  then
A12: f"{+infty} in S by A8,Th30;
A13: A /\ eq_dom(f,-infty) in S by A5,MESFUNC1:34;
  then reconsider B2 = f"{-infty} as Element of S by A10,Th30;
A14: f"{-infty} in S by A13,A10,Th30;
  thus f"{+infty} in S & f"{-infty} in S by A11,A13,A8,A10,Th30;
  set C2 = A \ B2;
A15: integral+(M,max-f) < +infty by A2;
  reconsider B1 = f"{+infty} as Element of S by A11,A8,Th30;
A16: A = dom max+f by A4,MESFUNC2:def 2;
  then
A17: B1 c= dom max+f by A7,Th30;
  then
A18: B1 = dom max+f /\ B1 by XBOOLE_1:28;
A19: max+f is A-measurable by A5,MESFUNC2:25;
  then max+f is B1-measurable by A16,A17,MESFUNC1:30;
  then
A20: max+f|B1 is B1-measurable by A18,Th42;
  set C1 = A \ B1;
A21: for x be Element of X holds ( x in dom(max+f|(B1\/C1)) implies max+f|(
  B1\/C1).x = max+f.x ) & ( x in dom(max-f|(B2\/C2)) implies max-f|(B2\/C2).x =
  max-f.x ) by FUNCT_1:47;
  B1\/C1 = A by A16,A17,XBOOLE_1:45;
  then dom(max+f|(B1\/C1)) = dom max+f /\ dom max+f by A16,RELAT_1:61;
  then max+f|(B1\/C1) = max+f by A21,PARTFUN1:5;
  then integral+(M,max+f) = integral+(M,max+f|B1) + integral+(M,max+f|C1) by A1
,A16,A19,Th81,XBOOLE_1:106;
  then
A22: integral+(M,max+f|B1) <= integral+(M,max+f) by A1,A16,A19,Th80,XXREAL_3:65
;
  thus now
A23: for r be Real st 0 < r holds r * M.B1 <= integral+(M,max+f)
    proof
      defpred P[object] means $1 in dom(max+f|B1);
      let r be Real;
      deffunc F(object) = In(r,ExtREAL);
A24:  for x be object st P[x] holds F(x) in ExtREAL;
      consider g be PartFunc of X,ExtREAL such that
A25:  (for x be object holds x in dom g iff x in X & P[x]) &
      for x be object st x in dom g holds g.x = F(x) from PARTFUN1:sch 3(A24);
      assume
A26:  0 < r;
      then for x be object st x in dom g holds 0 <= g.x by A25;
      then
A27:  g is nonnegative by SUPINF_2:52;
      dom(max+f|B1) = dom max+f /\ B1 by RELAT_1:61;
      then
A28:  dom(max+f|B1) = B1 by A17,XBOOLE_1:28;
      for x be object holds x in dom g iff x in X & x in dom(max+f|B1)
                by A25;
      then dom g = X /\ dom(max+f|B1) by XBOOLE_0:def 4;
      then
A29:  dom g = dom(max+f|B1) by XBOOLE_1:28;
      then
A30:  integral'(M,g) = r * M.(dom g) by A26,A25,A28,Th104;
A31:  for x be Element of X st x in dom g holds g.x <= max+f|B1.x
      proof
        let x be Element of X;
        assume
A32:    x in dom g;
        then x in dom f by A29,A28,FUNCT_1:def 7;
        then
A33:    x in dom max+f by MESFUNC2:def 2;
        f.x in {+infty} by A29,A28,A32,FUNCT_1:def 7;
        then
A34:    f.x = +infty by TARSKI:def 1;
        then max(f.x,0) = f.x by XXREAL_0:def 10;
        then max+f.x = +infty by A34,A33,MESFUNC2:def 2;
        then max+f|B1.x = +infty by A29,A28,A32,FUNCT_1:49;
        hence thesis by XXREAL_0:4;
      end;
      dom chi(B1,X) = X by FUNCT_3:def 3;
      then
A35:  B1 = dom chi(B1,X) /\ B1 by XBOOLE_1:28;
      then
A36:  chi(B1,X)|B1 is B1-measurable by Th42,MESFUNC2:29;
A37:  B1 = dom(chi(B1,X)|B1) by A35,RELAT_1:61;
A38:  for x be Element of X st x in dom g holds g.x = (r(#)(chi(B1,X)|B1) ).x
      proof
        let x be Element of X;
        assume
A39:    x in dom g;
        then x in dom(chi(B1,X)|B1) by A29,A28,A35,RELAT_1:61;
        then x in dom(r(#)(chi(B1,X)|B1)) by MESFUNC1:def 6;
        then
A40:    (r(#)(chi(B1,X)|B1)).x = r * (chi(B1,X)|B1).x by MESFUNC1:def 6
          .= r * chi(B1,X).x by A29,A28,A37,A39,FUNCT_1:47;
        chi(B1,X).x = 1 by A29,A28,A39,FUNCT_3:def 3;
        then (r(#)(chi(B1,X)|B1)).x = r by A40,XXREAL_3:81;
        hence thesis by A25,A39;
      end;
      dom g = dom(r(#)chi(B1,X)|B1) by A29,A28,A37,MESFUNC1:def 6;
      then g = r(#)(chi(B1,X)|B1) by A38,PARTFUN1:5;
      then
A41:  g is B1-measurable by A37,A36,MESFUNC1:37;
      max+f|B1 is nonnegative by Lm1,Th15;
      then
      integral+(M,g) <= integral+(M,max+f|B1) by A20,A29,A28,A41,A27,A31,Th85;
      then integral+(M,g) <= integral+(M,max+f) by A22,XXREAL_0:2;
      hence thesis by A25,A29,A28,A27,A30,Lm4,Th77;
    end;
    assume
A42: M.(f"{+infty}) <> 0;
    then
A43: 0 < M.(f"{+infty}) by A12,Th45;
    per cases;
    suppose
A44:  M.B1 = +infty;
      jj * M.B1 <= integral+(M,max+f) by A23;
      hence contradiction by A3,A44,XXREAL_3:81;
    end;
    suppose
      M.B1 <> +infty;
      then reconsider MB = M.B1 as Element of REAL by A43,XXREAL_0:14;
      jj * M.B1 <= integral+(M,max+f) by A23;
      then
A45:  0 < integral+(M,max+f) by A43;
      then reconsider I = integral+(M,max+ f) as Element of REAL
           by A3,XXREAL_0:14;
A46:  (2*I/MB) * M.B1 = 2*I/MB * MB;
      (2*I/MB) * M.B1 <= integral+(M,max+f) by A43,A23,A45;
      then 2 * I <= I by A42,A46,XCMPLX_1:87;
      hence contradiction by A45,XREAL_1:155;
    end;
  end;
  then reconsider B1 as measure_zero of M by MEASURE1:def 7;
A47: max-f is nonnegative by Lm1;
A48: A = dom max-f by A4,MESFUNC2:def 3;
  then
A49: B2 c= dom(max-f) by A9,Th30;
  then
A50: B2 = dom(max-f) /\ B2 by XBOOLE_1:28;
A51: max-f is A-measurable by A4,A5,MESFUNC2:26;
  then max-f is B2-measurable by A48,A49,MESFUNC1:30;
  then
A52: max-f|B2 is B2-measurable by A50,Th42;
  B2\/C2 = A by A48,A49,XBOOLE_1:45;
  then dom(max-f|(B2\/C2)) = dom max-f /\ dom max-f by A48,RELAT_1:61;
  then max-f|(B2\/C2) = max-f by A21,PARTFUN1:5;
  then integral+(M,max-f) = integral+(M,max-f|B2) + integral+(M,max-f|C2) by
A47,A48,A51,Th81,XBOOLE_1:106;
  then
A53: integral+(M,max-f|B2) <= integral+(M,max-f) by A47,A48,A51,Th80,
XXREAL_3:65;
  thus
A54: now
A55: for r be Real st 0 < r holds r * M.B2 <= integral+(M,max-f)
    proof
      defpred P[object] means $1 in dom(max-f|B2);
      let r be Real;
      deffunc F(object) = In(r,ExtREAL);
A56:  for x be object st P[x] holds F(x) in ExtREAL;
      consider g be PartFunc of X,ExtREAL such that
A57:  (for x be object holds x in dom g iff x in X & P[x]) & for x be
      object st x in dom g holds g.x = F(x) from PARTFUN1:sch 3(A56);
      assume
A58:  0 < r;
      then for x be object st x in dom g holds 0 <= g.x by A57;
      then
A59:  g is nonnegative by SUPINF_2:52;
      dom(max-f|B2) = dom max-f /\ B2 by RELAT_1:61;
      then
A60:  dom(max-f|B2) = B2 by A49,XBOOLE_1:28;
      for x be object holds x in dom g iff x in X & x in dom(max-f|B2) by A57;
      then dom g = X /\ dom(max-f|B2) by XBOOLE_0:def 4;
      then
A61:  dom g = dom(max- f|B2) by XBOOLE_1:28;
      then
A62:  integral'(M,g) = r * M.(dom g) by A58,A57,A60,Th104;
      dom chi(B2,X) = X by FUNCT_3:def 3;
      then
A63:  B2 = dom chi(B2,X) /\ B2 by XBOOLE_1:28;
      then
A64:  B2 = dom(chi(B2,X)|B2) by RELAT_1:61;
A65:  for x be Element of X st x in dom g holds g.x = (r(#)(chi(B2,X)|B2 )).x
      proof
        let x be Element of X;
        assume
A66:    x in dom g;
        then x in dom(r(#)(chi(B2,X)|B2)) by A61,A60,A64,MESFUNC1:def 6;
        then
A67:    (r(#)(chi(B2,X)|B2)).x = r * (chi(B2,X)|B2).x by MESFUNC1:def 6
          .= r * chi(B2,X).x by A61,A60,A64,A66,FUNCT_1:47;
        chi(B2,X).x = 1 by A61,A60,A66,FUNCT_3:def 3;
        then (r(#)(chi(B2,X)|B2)).x = r by A67,XXREAL_3:81;
        hence thesis by A57,A66;
      end;
A68:  for x be Element of X st x in dom g holds g.x <= (max-f|B2).x
      proof
        let x be Element of X;
        assume
A69:    x in dom g;
        then x in dom f by A61,A60,FUNCT_1:def 7;
        then
A70:    x in dom max-f by MESFUNC2:def 3;
        f.x in {-infty} by A61,A60,A69,FUNCT_1:def 7;
        then
A71:    -f.x = +infty by TARSKI:def 1,XXREAL_3:5;
        then max(-f.x,0) = -f.x by XXREAL_0:def 10;
        then max-f.x = +infty by A71,A70,MESFUNC2:def 3;
        then (max-f|B2).x = +infty by A61,A60,A69,FUNCT_1:49;
        hence thesis by XXREAL_0:4;
      end;
A72:  chi(B2,X)|B2 is B2-measurable by A63,Th42,MESFUNC2:29;
      dom g = dom(r(#)chi(B2,X)|B2) by A61,A60,A64,MESFUNC1:def 6;
      then g = r(#)(chi(B2,X)|B2) by A65,PARTFUN1:5;
      then
A73:  g is B2-measurable by A64,A72,MESFUNC1:37;
      max-f|B2 is nonnegative by Lm1,Th15;
      then integral+(M,g) <= integral+(M,max-f|B2) by A52,A61,A60,A73,A59,A68
,Th85;
      then integral+(M,g) <= integral+(M,max-f) by A53,XXREAL_0:2;
      hence thesis by A57,A61,A60,A59,A62,Lm4,Th77;
    end;
    assume
A74: M.(f"{-infty}) <> 0;
A75: 0 <= M.(f"{-infty}) by A14,Th45;
    per cases;
    suppose
A76:  M.B2 = +infty;
      jj * M.B2 <= integral+(M,max-f) by A55;
      hence contradiction by A15,A76,XXREAL_3:81;
    end;
    suppose
      M.B2 <> +infty;
      then reconsider MB = M.B2 as Element of REAL by A75,XXREAL_0:14;
      jj * M.B2 <= integral+(M,max-f) by A55;
      then
A77:  0 < integral+(M,max-f) by A74,A75;
      then reconsider I = integral+(M,max-f) as Element of REAL
            by A15,XXREAL_0:14;
A78:  (2*I/MB) * M.B2 = 2*I/MB * MB;
      (2*I/MB) * M.B2 <= integral+(M,max-f) by A74,A75,A55,A77;
      then 2 * I <= I by A74,A78,XCMPLX_1:87;
      hence contradiction by A77,XREAL_1:155;
    end;
  end;
  thus f"{+infty} \/ f"{-infty} in S by A12,A14,PROB_1:3;
  thus M.(f"{+infty} \/ f"{-infty}) = M.(B1 \/ B2) .= 0 by A54,MEASURE1:38;
end;
