let X be non empty set ; :: thesis: for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,ExtREAL
for E, A being Element of S st E = dom f & f is_measurable_on E & M . A = 0 holds
Integral M,(f | (E \ A)) = Integral M,f
let S be SigmaField of X; :: thesis: for M being sigma_Measure of S
for f being PartFunc of X,ExtREAL
for E, A being Element of S st E = dom f & f is_measurable_on E & M . A = 0 holds
Integral M,(f | (E \ A)) = Integral M,f
let M be sigma_Measure of S; :: thesis: for f being PartFunc of X,ExtREAL
for E, A being Element of S st E = dom f & f is_measurable_on E & M . A = 0 holds
Integral M,(f | (E \ A)) = Integral M,f
let f be PartFunc of X,ExtREAL ; :: thesis: for E, A being Element of S st E = dom f & f is_measurable_on E & M . A = 0 holds
Integral M,(f | (E \ A)) = Integral M,f
let E, A be Element of S; :: thesis: ( E = dom f & f is_measurable_on E & M . A = 0 implies Integral M,(f | (E \ A)) = Integral M,f )
assume A1: ( E = dom f & f is_measurable_on E & M . A = 0 ) ; :: thesis: Integral M,(f | (E \ A)) = Integral M,f
set B = E \ A;
A2: ( dom f = dom (max+ f) & dom f = dom (max- f) ) by MESFUNC2:def 2, MESFUNC2:def 3;
A3: ( max+ f is nonnegative & max- f is nonnegative ) by Lm1;
A4: ( max+ f is_measurable_on E & max- f is_measurable_on E ) by A1, MESFUNC2:27, MESFUNC2:28;
Integral M,(f | (E \ A)) = (integral+ M,((max+ f) | (E \ A))) - (integral+ M,(max- (f | (E \ A)))) by Th34
.= (integral+ M,((max+ f) | (E \ A))) - (integral+ M,((max- f) | (E \ A))) by Th34
.= (integral+ M,(max+ f)) - (integral+ M,((max- f) | (E \ A))) by A1, A2, A3, A4, Th90 ;
hence Integral M,(f | (E \ A)) = Integral M,f by A1, A2, A3, A4, Th90; :: thesis: verum