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,REAL
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,REAL
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,REAL
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,REAL ; :: 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 A1: ( ex E being Element of S st
( E = dom f & f is_measurable_on E ) & f is nonnegative & A c= B ) ; :: thesis: Integral M,(f | A) <= Integral M,(f | B)
then consider E being Element of S such that
A2: ( E = dom f & f is_measurable_on E ) ;
( E = dom (R_EAL f) & R_EAL f is_measurable_on E ) by A2, Def6;
hence Integral M,(f | A) <= Integral M,(f | B) by A1, MESFUNC5:99; :: thesis: verum