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_measurable_on E ) & 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_measurable_on E ) & 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_measurable_on E ) & 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_measurable_on E ) & 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_measurable_on E ) & 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_measurable_on E ) 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, Th87, XBOOLE_1:89;
hence integral+ M,(f | A) <= integral+ M,(f | B) by A1, A2, A4, A5, Th86, XXREAL_3:42; :: thesis: verum