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 being Element of S st ex E being Element of S st
( E = dom f & f is_measurable_on E ) & M . A = 0 holds
Integral (M,(f | A)) = 0
let S be SigmaField of X; :: thesis: for M being sigma_Measure of S
for f being PartFunc of X,REAL
for A being Element of S st ex E being Element of S st
( E = dom f & f is_measurable_on E ) & M . A = 0 holds
Integral (M,(f | A)) = 0
let M be sigma_Measure of S; :: thesis: for f being PartFunc of X,REAL
for A being Element of S st ex E being Element of S st
( E = dom f & f is_measurable_on E ) & M . A = 0 holds
Integral (M,(f | A)) = 0
let f be PartFunc of X,REAL; :: thesis: for A being Element of S st ex E being Element of S st
( E = dom f & f is_measurable_on E ) & M . A = 0 holds
Integral (M,(f | A)) = 0
let A be Element of S; :: thesis: ( ex E being Element of S st
( E = dom f & f is_measurable_on E ) & M . A = 0 implies Integral (M,(f | A)) = 0 )
assume that
A1: ex E being Element of S st
( E = dom f & f is_measurable_on E ) and
A2: M . A = 0 ; :: thesis: Integral (M,(f | A)) = 0
consider E being Element of S such that
A3: E = dom f and
A4: f is_measurable_on E by A1;
R_EAL f is_measurable_on E by A4, Def6;
hence Integral (M,(f | A)) = 0 by A2, A3, MESFUNC5:94; :: thesis: verum