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 MEASURE1:21;
set F = XX \ E;

A1: now

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

per cases ( 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_1:90;

A7: 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 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_3:43;
then (max+ (chi E,X)) . x = 0 by MESFUNC7:23;
hence ((max+ (chi E,X)) | (XX \ E)) . x = 0 by A8, FUNCT_1:70; :: thesis: verum

end;

A9: chi E,X is_measurable_on XX by MESFUNC2:32;
then A10: max+ (chi E,X) is_measurable_on XX by MESFUNC7:23;
then max+ (chi E,X) is_measurable_on XX \ E by MESFUNC1:34;
then A11: integral+ M,((max+ (chi E,X)) | (XX \ E)) = 0 by A4, A5, A6, A7, MESFUNC5:48, MESFUNC5:93;
A12: XX /\ E = E by XBOOLE_1:28;
then A13: dom ((max+ (chi E,X)) | E) = E by A4, RELAT_1:90;
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_1:98;
A15: for x being set st x in dom (max+ (chi E,X)) holds
0. <= (max+ (chi E,X)) . x by MESFUNC2:14;
then A16: max+ (chi E,X) is nonnegative by SUPINF_2:71;
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, MESFUNC5:87, 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

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_1:70; :: 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, MESFUNC5:21, MESFUNC5:83;
then A20: integral+ M,(max+ (chi E,X)) = (R_EAL 1) * (M . (dom ((max+ (chi E,X)) | E))) by A13, A18, MESFUNC5:110;
max+ (chi E,X) is_measurable_on E by A10, MESFUNC1:34;
then (max+ (chi E,X)) | E is_measurable_on E by A4, A12, MESFUNC5:48;
then A21: (chi E,X) | E is_measurable_on E by MESFUNC7:23;
(max+ (chi E,X)) | E is nonnegative by A15, MESFUNC5:21, SUPINF_2:71;
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, MESFUNC5:94;
XX = dom (max- (chi E,X)) by A3, MESFUNC2:def 3;
then integral+ M,(max- (chi E,X)) = 0 by A3, A9, A1, MESFUNC2:28, MESFUNC5:93;
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_3:92; :: 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_3:92; :: thesis: verum