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))

set A9 = A /\ B;

A4: A /\ B = A by A3, XBOOLE_1:28;

set B9 = B \ A;

A5: (A /\ B) \/ (B \ A) = B by XBOOLE_1:51;

integral+ (M,(f | ((A /\ B) \/ (B \ A)))) = (integral+ (M,(f | (A /\ B)))) + (integral+ (M,(f | (B \ A)))) by A1, A2, Th81, XBOOLE_1:89;

hence integral+ (M,(f | A)) <= integral+ (M,(f | B)) by A1, A2, A4, A5, Th80, XXREAL_3:39; :: 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))

set A9 = A /\ B;

A4: A /\ B = A by A3, XBOOLE_1:28;

set B9 = B \ A;

A5: (A /\ B) \/ (B \ A) = B by XBOOLE_1:51;

integral+ (M,(f | ((A /\ B) \/ (B \ A)))) = (integral+ (M,(f | (A /\ B)))) + (integral+ (M,(f | (B \ A)))) by A1, A2, Th81, XBOOLE_1:89;

hence integral+ (M,(f | A)) <= integral+ (M,(f | B)) by A1, A2, A4, A5, Th80, XXREAL_3:39; :: thesis: verum