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

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