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 that
A1: E = dom f and
A2: f is_measurable_on E and
A3: M . A = 0 ; :: thesis: Integral (M,(f | (E \ A))) = Integral (M,f)
R_EAL f is_measurable_on E by A2, Def6;
hence Integral (M,(f | (E \ A))) = Integral (M,f) by A1, A3, MESFUNC5:95; :: thesis: verum