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 A being Element of S st ex E being Element of S st
( E = dom f & f is_measurable_on E ) & M . A = 0 holds
Integral M,(f | A) = 0
let S be SigmaField of X; :: thesis: for M being sigma_Measure of S
for f being PartFunc of X,ExtREAL
for A being Element of S st ex E being Element of S st
( E = dom f & f is_measurable_on E ) & M . A = 0 holds
Integral M,(f | A) = 0
let M be sigma_Measure of S; :: thesis: for f being PartFunc of X,ExtREAL
for A being Element of S st ex E being Element of S st
( E = dom f & f is_measurable_on E ) & M . A = 0 holds
Integral M,(f | A) = 0
let f be PartFunc of X,ExtREAL ; :: thesis: for A being Element of S st ex E being Element of S st
( E = dom f & f is_measurable_on E ) & M . A = 0 holds
Integral M,(f | A) = 0
let A be Element of S; :: thesis: ( ex E being Element of S st
( E = dom f & f is_measurable_on E ) & M . A = 0 implies Integral M,(f | A) = 0 )
assume A1: ( ex E being Element of S st
( E = dom f & f is_measurable_on E ) & M . A = 0 ) ; :: thesis: Integral M,(f | A) = 0
consider E being Element of S;
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;
then A4: integral+ M,((max+ f) | A) = 0 by A1, A2, Th88, MESFUNC2:27;
Integral M,(f | A) = (integral+ M,((max+ f) | A)) - (integral+ M,(max- (f | A))) by Th34
.= (integral+ M,((max+ f) | A)) - (integral+ M,((max- f) | A)) by Th34
.= 0. - 0. by A1, A2, A3, A4, Th88, MESFUNC2:28 ;
hence Integral M,(f | A) = 0 ; :: thesis: verum