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 E -measurable ) & 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 E -measurable ) & 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 E -measurable ) & 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 E -measurable ) & 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 E -measurable ) & M . A = 0 implies Integral (M,(f | A)) = 0 )
assume that
A1: ex E being Element of S st
( E = dom f & f is E -measurable ) and
A2: M . A = 0 ; :: thesis: Integral (M,(f | A)) = 0
A3: dom f = dom (max+ f) by MESFUNC2:def 2;
max+ f is nonnegative by Lm1;
then A4: integral+ (M,((max+ f) | A)) = 0 by A1, A2, A3, Th82, MESFUNC2:25;
A5: dom f = dom (max- f) by MESFUNC2:def 3;
A6: max- f is nonnegative by Lm1;
Integral (M,(f | A)) = (integral+ (M,((max+ f) | A))) - (integral+ (M,(max- (f | A)))) by Th28
.= (integral+ (M,((max+ f) | A))) - (integral+ (M,((max- f) | A))) by Th28
.= 0. - 0. by A1, A2, A5, A6, A4, Th82, MESFUNC2:26 ;
hence Integral (M,(f | A)) = 0 ; :: thesis: verum