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 E, A being Element of S st E = dom f & f is_measurable_on E & M . A = 0 holds
Integral M,(f | (E \ A)) = Integral M,f
let S be SigmaField of X; :: thesis: for M being sigma_Measure of S
for f being PartFunc of X,REAL
for E, A being Element of S st E = dom f & f is_measurable_on E & M . A = 0 holds
Integral M,(f | (E \ A)) = Integral M,f
let M be sigma_Measure of S; :: thesis: for f being PartFunc of X,REAL
for E, A being Element of S st E = dom f & f is_measurable_on E & M . A = 0 holds
Integral M,(f | (E \ A)) = Integral M,f
let f be PartFunc of X,REAL ; :: thesis: for E, A being Element of S st E = dom f & f is_measurable_on E & M . A = 0 holds
Integral M,(f | (E \ A)) = Integral M,f
let E, A be Element of S; :: thesis: ( E = dom f & f is_measurable_on E & M . A = 0 implies Integral M,(f | (E \ A)) = Integral M,f )
assume A1: ( E = dom f & f is_measurable_on E & M . A = 0 ) ; :: thesis: Integral M,(f | (E \ A)) = Integral M,f
then ( E = dom (R_EAL f) & R_EAL f is_measurable_on E ) by Def6;
hence Integral M,(f | (E \ A)) = Integral M,f by A1, MESFUNC5:101; :: thesis: verum