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 misses B holds
Integral (M,(f | (A \/ B))) = (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 misses B holds
Integral (M,(f | (A \/ B))) = (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 misses B holds
Integral (M,(f | (A \/ B))) = (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 misses B holds
Integral (M,(f | (A \/ B))) = (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 misses B implies Integral (M,(f | (A \/ B))) = (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 & A misses B ) ; :: thesis: Integral (M,(f | (A \/ B))) = (Integral (M,(f | A))) + (Integral (M,(f | B)))
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 \/ B))) = (Integral (M,(f | A))) + (Integral (M,(f | B))) by A2, A3, MESFUNC5:91; :: thesis: verum