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 ) )
assume A1: 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 )
reconsider XX = X as Element of S by MEASURE1:21;
A2: ( XX = dom (chi E,X) & chi E,X is_measurable_on XX ) by FUNCT_3:def 3, MESFUNC2:32;
then A3: ( XX = dom (max+ (chi E,X)) & max+ (chi E,X) is_measurable_on XX ) by Th23;
set F = XX \ E;
for x being set st x in dom (max+ (chi E,X)) holds
0. <= (max+ (chi E,X)) . x by MESFUNC2:14;
then A4: max+ (chi E,X) is nonnegative by SUPINF_2:71;
then A5: (max+ (chi E,X)) | E is nonnegative by MESFUNC5:21;
A6: 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 A3, A4, MESFUNC5:87, XBOOLE_1:79;
A7: ( XX /\ (XX \ E) = XX \ E & XX /\ E = E ) by XBOOLE_1:28;
then A8: ( dom ((max+ (chi E,X)) | (XX \ E)) = XX \ E & dom ((max+ (chi E,X)) | E) = E ) by A3, RELAT_1:90;
( max+ (chi E,X) is_measurable_on E & max+ (chi E,X) is_measurable_on XX \ E ) by A3, MESFUNC1:34;
then A9: ( (max+ (chi E,X)) | E is_measurable_on E & (max+ (chi E,X)) | (XX \ E) is_measurable_on XX \ E ) by A3, A7, MESFUNC5:48;
E \/ (XX \ E) = XX by A7, XBOOLE_1:51;
then A10: (max+ (chi E,X)) | (E \/ (XX \ E)) = max+ (chi E,X) by A3, RELAT_1:98;

now

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 A11: 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 A8, FUNCT_3:43;
then (max+ (chi E,X)) . x = 0 by Th23;
hence ((max+ (chi E,X)) | (XX \ E)) . x = 0 by A11, FUNCT_1:70; :: thesis: verum

end;

then integral+ M,((max+ (chi E,X)) | (XX \ E)) = 0 by A8, A9, MESFUNC5:93;
then A12: integral+ M,(max+ (chi E,X)) = integral+ M,((max+ (chi E,X)) | E) by A6, A10, XXREAL_3:4;

A13: now

let x be set ; :: thesis: ( x in dom ((max+ (chi E,X)) | E) implies ((max+ (chi E,X)) | E) . x = 1 )
assume A14: x in dom ((max+ (chi E,X)) | E) ; :: thesis: ((max+ (chi E,X)) | E) . x = 1
then (chi E,X) . x = 1 by A8, FUNCT_3:def 3;
then (max+ (chi E,X)) . x = 1 by Th23;
hence ((max+ (chi E,X)) | E) . x = 1 by A14, FUNCT_1:70; :: thesis: verum

end;

then (max+ (chi E,X)) | E is_simple_func_in S by A8, MESFUNC6:2;
then integral+ M,(max+ (chi E,X)) = integral' M,((max+ (chi E,X)) | E) by A4, A12, MESFUNC5:21, MESFUNC5:83;
then A15: integral+ M,(max+ (chi E,X)) = (R_EAL 1) * (M . (dom ((max+ (chi E,X)) | E))) by A8, A13, MESFUNC5:110;
then A16: integral+ M,(max+ (chi E,X)) < +infty by A1, A8, XXREAL_3:92;
A17: ( XX = dom (max- (chi E,X)) & max- (chi E,X) is_measurable_on XX ) by A2, MESFUNC2:28, MESFUNC2:def 3;

now

let x be Element of X; :: thesis: ( x in dom (max- (chi E,X)) implies (max- (chi E,X)) . x = 0 )
assume A18: x in dom (max- (chi E,X)) ; :: thesis: (max- (chi E,X)) . x = 0

A19: now

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;

now

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;

hence (max- (chi E,X)) . x = 0 by A18, A19, MESFUNC2:def 3; :: thesis: verum

end;

then A21: integral+ M,(max- (chi E,X)) = 0 by A17, MESFUNC5:93;
hence chi E,X is_integrable_on M by A2, A16, MESFUNC5:def 17; :: thesis: ( Integral M,(chi E,X) = M . E & Integral M,((chi E,X) | E) = M . E )
Integral M,(chi E,X) = (R_EAL 1) * (M . E) by A8, A15, A21, XXREAL_3:15;
hence Integral M,(chi E,X) = M . E by XXREAL_3:92; :: thesis: Integral M,((chi E,X) | E) = M . E
A22: ( E = dom ((chi E,X) | E) & (chi E,X) | E is_measurable_on E ) by A8, A9, Th23;
A23: (chi E,X) | E is nonnegative by A5, Th23;
A24: (chi E,X) | E = (max+ (chi E,X)) | E by Th23;
Integral M,((chi E,X) | E) = integral+ M,((chi E,X) | E) by A22, A23, MESFUNC5:94;
hence Integral M,((chi E,X) | E) = M . E by A8, A12, A15, A24, XXREAL_3:92; :: thesis: verum