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

f | A is_integrable_on M

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

f | A is_integrable_on M

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

f | A is_integrable_on M

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

f | A is_integrable_on M

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 f | A is_integrable_on M )

assume that

A1: ex E being Element of S st

( E = dom f & f is E -measurable ) and

A2: M . A = 0 ; :: thesis: f | A is_integrable_on M

A3: dom f = dom (max+ f) by MESFUNC2:def 2;

max+ f is nonnegative by Lm1;

then integral+ (M,((max+ f) | A)) = 0 by A1, A2, A3, MESFUNC2:25, MESFUNC5:82;

then A4: integral+ (M,(max+ (f | A))) < +infty by MESFUNC5:28;

consider E being Element of S such that

A5: E = dom f and

A6: f is E -measurable by A1;

A7: (dom f) /\ (A /\ E) = A /\ E by A5, XBOOLE_1:17, XBOOLE_1:28;

A8: dom f = dom (max- f) by MESFUNC2:def 3;

max- f is nonnegative by Lm1;

then integral+ (M,((max- f) | A)) = 0 by A1, A2, A8, MESFUNC2:26, MESFUNC5:82;

then A9: integral+ (M,(max- (f | A))) < +infty by MESFUNC5:28;

A10: dom (f | A) = (dom f) /\ A by RELAT_1:61;

f is A /\ E -measurable by A6, MESFUNC1:30, XBOOLE_1:17;

then A11: f | (A /\ E) is A /\ E -measurable by A7, MESFUNC5:42;

f | (A /\ E) = (f | A) /\ (f | E) by RELAT_1:79

.= (f | A) /\ f by A5

.= f | A by RELAT_1:59, XBOOLE_1:28 ;

hence f | A is_integrable_on M by A5, A11, A10, A4, A9; :: 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

f | A is_integrable_on M

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

f | A is_integrable_on M

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

f | A is_integrable_on M

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

f | A is_integrable_on M

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 f | A is_integrable_on M )

assume that

A1: ex E being Element of S st

( E = dom f & f is E -measurable ) and

A2: M . A = 0 ; :: thesis: f | A is_integrable_on M

A3: dom f = dom (max+ f) by MESFUNC2:def 2;

max+ f is nonnegative by Lm1;

then integral+ (M,((max+ f) | A)) = 0 by A1, A2, A3, MESFUNC2:25, MESFUNC5:82;

then A4: integral+ (M,(max+ (f | A))) < +infty by MESFUNC5:28;

consider E being Element of S such that

A5: E = dom f and

A6: f is E -measurable by A1;

A7: (dom f) /\ (A /\ E) = A /\ E by A5, XBOOLE_1:17, XBOOLE_1:28;

A8: dom f = dom (max- f) by MESFUNC2:def 3;

max- f is nonnegative by Lm1;

then integral+ (M,((max- f) | A)) = 0 by A1, A2, A8, MESFUNC2:26, MESFUNC5:82;

then A9: integral+ (M,(max- (f | A))) < +infty by MESFUNC5:28;

A10: dom (f | A) = (dom f) /\ A by RELAT_1:61;

f is A /\ E -measurable by A6, MESFUNC1:30, XBOOLE_1:17;

then A11: f | (A /\ E) is A /\ E -measurable by A7, MESFUNC5:42;

f | (A /\ E) = (f | A) /\ (f | E) by RELAT_1:79

.= (f | A) /\ f by A5

.= f | A by RELAT_1:59, XBOOLE_1:28 ;

hence f | A is_integrable_on M by A5, A11, A10, A4, A9; :: thesis: verum