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 st M . E < +infty holds
( chi (E,X) is_integrable_on M & 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 st M . E < +infty holds
( chi (E,X) is_integrable_on M & 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 st M . E < +infty holds
( chi (E,X) is_integrable_on M & Integral (M,(chi (E,X))) = M . E & Integral (M,((chi (E,X)) | E)) = M . E )
let E be Element of S; :: thesis: ( M . E < +infty implies ( chi (E,X) is_integrable_on M & 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))) . x = 0 )
A2: now :: thesis: ( x in E implies max ((- ((chi (E,X)) . x)),0.) = 0. )
assume x in E ; :: thesis: max ((- ((chi (E,X)) . x)),0.) = 0.
then (chi (E,X)) . x = 1 by FUNCT_3:def 3;
hence max ((- ((chi (E,X)) . x)),0.) = 0. by XXREAL_0:def 10; :: thesis: verum
end;
A3: now :: thesis: ( not x in E implies max ((- ((chi (E,X)) . x)),0.) = 0. )
assume not x in E ; :: thesis: max ((- ((chi (E,X)) . x)),0.) = 0.
then (chi (E,X)) . x = 0. by FUNCT_3:def 3;
then - ((chi (E,X)) . x) = 0 ;
hence max ((- ((chi (E,X)) . x)),0.) = 0. ; :: thesis: verum
end;
assume x in dom (max- (chi (E,X))) ; :: thesis: (max- (chi (E,X))) . x = 0
hence (max- (chi (E,X))) . x = 0 by A2, A3, MESFUNC2:def 3; :: thesis: verum
end;
A4: XX = dom (chi (E,X)) by FUNCT_3:def 3;
then A5: XX = dom (max+ (chi (E,X))) by Th23;
A6: XX /\ (XX \ E) = XX \ E by XBOOLE_1:28;
then A7: dom ((max+ (chi (E,X))) | (XX \ E)) = XX \ E by A5, RELAT_1:61;
A8: 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 A9: 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 A7, FUNCT_3:37;
then (max+ (chi (E,X))) . x = 0 by Th23;
hence ((max+ (chi (E,X))) | (XX \ E)) . x = 0 by A9, FUNCT_1:47; :: thesis: verum
end;
A10: chi (E,X) is XX -measurable by MESFUNC2:29;
then A11: max+ (chi (E,X)) is XX -measurable by Th23;
then max+ (chi (E,X)) is XX \ E -measurable by MESFUNC1:30;
then A12: integral+ (M,((max+ (chi (E,X))) | (XX \ E))) = 0 by A5, A6, A7, A8, MESFUNC5:42, MESFUNC5:87;
XX = dom (max- (chi (E,X))) by A4, MESFUNC2:def 3;
then A13: integral+ (M,(max- (chi (E,X)))) = 0 by A4, A10, A1, MESFUNC2:26, MESFUNC5:87;
A14: XX /\ E = E by XBOOLE_1:28;
then A15: dom ((max+ (chi (E,X))) | E) = E by A5, RELAT_1:61;
E \/ (XX \ E) = XX by A14, XBOOLE_1:51;
then A16: (max+ (chi (E,X))) | (E \/ (XX \ E)) = max+ (chi (E,X)) by A5, RELAT_1:69;
A17: for x being object st x in dom (max+ (chi (E,X))) holds
0. <= (max+ (chi (E,X))) . x by MESFUNC2:12;
then A18: 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 A5, A11, MESFUNC5:81, XBOOLE_1:79;
then A19: integral+ (M,(max+ (chi (E,X)))) = integral+ (M,((max+ (chi (E,X))) | E)) by A16, A12, XXREAL_3:4;
A20: now :: thesis: for x being object st x in dom ((max+ (chi (E,X))) | E) holds
((max+ (chi (E,X))) | E) . x = jj
let x be object ; :: thesis: ( x in dom ((max+ (chi (E,X))) | E) implies ((max+ (chi (E,X))) | E) . x = jj )
assume A21: x in dom ((max+ (chi (E,X))) | E) ; :: thesis: ((max+ (chi (E,X))) | E) . x = jj
then (chi (E,X)) . x = 1 by A15, FUNCT_3:def 3;
then (max+ (chi (E,X))) . x = 1 by Th23;
hence ((max+ (chi (E,X))) | E) . x = jj by A21, FUNCT_1:47; :: thesis: verum
end;
then (max+ (chi (E,X))) | E is_simple_func_in S by A15, MESFUNC6:2;
then integral+ (M,(max+ (chi (E,X)))) = integral' (M,((max+ (chi (E,X))) | E)) by A18, A19, MESFUNC5:15, MESFUNC5:77;
then A22: integral+ (M,(max+ (chi (E,X)))) = jj * (M . (dom ((max+ (chi (E,X))) | E))) by A15, A20, MESFUNC5:104;
max+ (chi (E,X)) is E -measurable by A11, MESFUNC1:30;
then (max+ (chi (E,X))) | E is E -measurable by A5, A14, MESFUNC5:42;
then A23: (chi (E,X)) | E is E -measurable by Th23;
(max+ (chi (E,X))) | E is nonnegative by A17, MESFUNC5:15, SUPINF_2:52;
then A24: (chi (E,X)) | E is nonnegative by Th23;
E = dom ((chi (E,X)) | E) by A15, Th23;
then A25: Integral (M,((chi (E,X)) | E)) = integral+ (M,((chi (E,X)) | E)) by A23, A24, MESFUNC5:88;
assume M . E < +infty ; :: thesis: ( chi (E,X) is_integrable_on M & Integral (M,(chi (E,X))) = M . E & Integral (M,((chi (E,X)) | E)) = M . E )
then integral+ (M,(max+ (chi (E,X)))) < +infty by A15, A22, XXREAL_3:81;
hence chi (E,X) is_integrable_on M by A4, A10, A13; :: thesis: ( Integral (M,(chi (E,X))) = M . E & Integral (M,((chi (E,X)) | E)) = M . E )
Integral (M,(chi (E,X))) = 1 * (M . E) by A15, A22, A13, 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 Th23;
hence Integral (M,((chi (E,X)) | E)) = M . E by A15, A19, A22, A25, XXREAL_3:81; :: thesis: verum