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, B being Element of S st ex E being Element of S st

( E = dom f & f is E -measurable ) & f is nonnegative & A c= B holds

Integral (M,(f | A)) <= Integral (M,(f | B))

let S be SigmaField of X; :: thesis: for M being sigma_Measure of S

for f being PartFunc of X,ExtREAL

for A, B being Element of S st ex E being Element of S st

( E = dom f & f is E -measurable ) & f is nonnegative & A c= B holds

Integral (M,(f | A)) <= Integral (M,(f | B))

let M be sigma_Measure of S; :: thesis: for f being PartFunc of X,ExtREAL

for A, B being Element of S st ex E being Element of S st

( E = dom f & f is E -measurable ) & f is nonnegative & A c= B holds

Integral (M,(f | A)) <= Integral (M,(f | B))

let f be PartFunc of X,ExtREAL; :: thesis: for A, B being Element of S st ex E being Element of S st

( E = dom f & f is E -measurable ) & f is nonnegative & A c= B holds

Integral (M,(f | A)) <= Integral (M,(f | B))

let A, B be Element of S; :: thesis: ( ex E being Element of S st

( E = dom f & f is E -measurable ) & f is nonnegative & A c= B implies Integral (M,(f | A)) <= Integral (M,(f | B)) )

assume that

A1: ex E being Element of S st

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

A2: f is nonnegative and

A3: A c= B ; :: thesis: Integral (M,(f | A)) <= Integral (M,(f | B))

consider E being Element of S such that

A4: E = dom f and

A5: f is E -measurable by A1;

A6: ex C being Element of S st

( C = dom (f | A) & f | A is C -measurable )

( C = dom (f | B) & f | B is C -measurable )

then Integral (M,(f | A)) <= integral+ (M,(f | B)) by A2, A6, Th15, Th88;

hence Integral (M,(f | A)) <= Integral (M,(f | B)) by A2, A11, Th15, Th88; :: thesis: verum

for M being sigma_Measure of S

for f being PartFunc of X,ExtREAL

for A, B being Element of S st ex E being Element of S st

( E = dom f & f is E -measurable ) & f is nonnegative & A c= B holds

Integral (M,(f | A)) <= Integral (M,(f | B))

let S be SigmaField of X; :: thesis: for M being sigma_Measure of S

for f being PartFunc of X,ExtREAL

for A, B being Element of S st ex E being Element of S st

( E = dom f & f is E -measurable ) & f is nonnegative & A c= B holds

Integral (M,(f | A)) <= Integral (M,(f | B))

let M be sigma_Measure of S; :: thesis: for f being PartFunc of X,ExtREAL

for A, B being Element of S st ex E being Element of S st

( E = dom f & f is E -measurable ) & f is nonnegative & A c= B holds

Integral (M,(f | A)) <= Integral (M,(f | B))

let f be PartFunc of X,ExtREAL; :: thesis: for A, B being Element of S st ex E being Element of S st

( E = dom f & f is E -measurable ) & f is nonnegative & A c= B holds

Integral (M,(f | A)) <= Integral (M,(f | B))

let A, B be Element of S; :: thesis: ( ex E being Element of S st

( E = dom f & f is E -measurable ) & f is nonnegative & A c= B implies Integral (M,(f | A)) <= Integral (M,(f | B)) )

assume that

A1: ex E being Element of S st

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

A2: f is nonnegative and

A3: A c= B ; :: thesis: Integral (M,(f | A)) <= Integral (M,(f | B))

consider E being Element of S such that

A4: E = dom f and

A5: f is E -measurable by A1;

A6: ex C being Element of S st

( C = dom (f | A) & f | A is C -measurable )

proof

A11:
ex C being Element of S st
take C = E /\ A; :: thesis: ( C = dom (f | A) & f | A is C -measurable )

thus dom (f | A) = C by A4, RELAT_1:61; :: thesis: f | A is C -measurable

A7: C = (dom f) /\ C by A4, XBOOLE_1:17, XBOOLE_1:28;

A8: dom (f | A) = C by A4, RELAT_1:61

.= dom (f | C) by A7, RELAT_1:61 ;

A9: for x being object st x in dom (f | A) holds

(f | A) . x = (f | C) . x

then f | C is C -measurable by A7, Th42;

hence f | A is C -measurable by A8, A9, FUNCT_1:2; :: thesis: verum

end;thus dom (f | A) = C by A4, RELAT_1:61; :: thesis: f | A is C -measurable

A7: C = (dom f) /\ C by A4, XBOOLE_1:17, XBOOLE_1:28;

A8: dom (f | A) = C by A4, RELAT_1:61

.= dom (f | C) by A7, RELAT_1:61 ;

A9: for x being object st x in dom (f | A) holds

(f | A) . x = (f | C) . x

proof

f is C -measurable
by A5, MESFUNC1:30, XBOOLE_1:17;
let x be object ; :: thesis: ( x in dom (f | A) implies (f | A) . x = (f | C) . x )

assume A10: x in dom (f | A) ; :: thesis: (f | A) . x = (f | C) . x

then (f | A) . x = f . x by FUNCT_1:47;

hence (f | A) . x = (f | C) . x by A8, A10, FUNCT_1:47; :: thesis: verum

end;assume A10: x in dom (f | A) ; :: thesis: (f | A) . x = (f | C) . x

then (f | A) . x = f . x by FUNCT_1:47;

hence (f | A) . x = (f | C) . x by A8, A10, FUNCT_1:47; :: thesis: verum

then f | C is C -measurable by A7, Th42;

hence f | A is C -measurable by A8, A9, FUNCT_1:2; :: thesis: verum

( C = dom (f | B) & f | B is C -measurable )

proof

integral+ (M,(f | A)) <= integral+ (M,(f | B))
by A1, A2, A3, Th83;
take C = E /\ B; :: thesis: ( C = dom (f | B) & f | B is C -measurable )

thus dom (f | B) = C by A4, RELAT_1:61; :: thesis: f | B is C -measurable

A12: C = (dom f) /\ C by A4, XBOOLE_1:17, XBOOLE_1:28;

A13: dom (f | B) = C by A4, RELAT_1:61

.= dom (f | C) by A12, RELAT_1:61 ;

A14: for x being object st x in dom (f | B) holds

(f | B) . x = (f | C) . x

then f | C is C -measurable by A12, Th42;

hence f | B is C -measurable by A13, A14, FUNCT_1:2; :: thesis: verum

end;thus dom (f | B) = C by A4, RELAT_1:61; :: thesis: f | B is C -measurable

A12: C = (dom f) /\ C by A4, XBOOLE_1:17, XBOOLE_1:28;

A13: dom (f | B) = C by A4, RELAT_1:61

.= dom (f | C) by A12, RELAT_1:61 ;

A14: for x being object st x in dom (f | B) holds

(f | B) . x = (f | C) . x

proof

f is C -measurable
by A5, MESFUNC1:30, XBOOLE_1:17;
let x be object ; :: thesis: ( x in dom (f | B) implies (f | B) . x = (f | C) . x )

assume A15: x in dom (f | B) ; :: thesis: (f | B) . x = (f | C) . x

then (f | B) . x = f . x by FUNCT_1:47;

hence (f | B) . x = (f | C) . x by A13, A15, FUNCT_1:47; :: thesis: verum

end;assume A15: x in dom (f | B) ; :: thesis: (f | B) . x = (f | C) . x

then (f | B) . x = f . x by FUNCT_1:47;

hence (f | B) . x = (f | C) . x by A13, A15, FUNCT_1:47; :: thesis: verum

then f | C is C -measurable by A12, Th42;

hence f | B is C -measurable by A13, A14, FUNCT_1:2; :: thesis: verum

then Integral (M,(f | A)) <= integral+ (M,(f | B)) by A2, A6, Th15, Th88;

hence Integral (M,(f | A)) <= Integral (M,(f | B)) by A2, A11, Th15, Th88; :: thesis: verum