let X be non empty set ; :: thesis: for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,REAL st f is_integrable_on M & g is_integrable_on M holds
ex E being Element of S st
( E = (dom f) /\ (dom g) & Integral M,(f - g) = (Integral M,(f | E)) + (Integral M,((- g) | E)) )
let S be SigmaField of X; :: thesis: for M being sigma_Measure of S
for f, g being PartFunc of X,REAL st f is_integrable_on M & g is_integrable_on M holds
ex E being Element of S st
( E = (dom f) /\ (dom g) & Integral M,(f - g) = (Integral M,(f | E)) + (Integral M,((- g) | E)) )
let M be sigma_Measure of S; :: thesis: for f, g being PartFunc of X,REAL st f is_integrable_on M & g is_integrable_on M holds
ex E being Element of S st
( E = (dom f) /\ (dom g) & Integral M,(f - g) = (Integral M,(f | E)) + (Integral M,((- g) | E)) )
let f, g be PartFunc of X,REAL ; :: thesis: ( f is_integrable_on M & g is_integrable_on M implies ex E being Element of S st
( E = (dom f) /\ (dom g) & Integral M,(f - g) = (Integral M,(f | E)) + (Integral M,((- g) | E)) ) )
assume that
A1: f is_integrable_on M and
A2: g is_integrable_on M ; :: thesis: ex E being Element of S st
( E = (dom f) /\ (dom g) & Integral M,(f - g) = (Integral M,(f | E)) + (Integral M,((- g) | E)) )
R_EAL g is_integrable_on M by A2, MESFUNC6:def 9;
then (- 1) (#) (R_EAL g) is_integrable_on M by MESFUNC5:116;
then - (R_EAL g) is_integrable_on M by MESFUNC2:11;
then A3: R_EAL (- g) is_integrable_on M by MESFUNC6:28;
R_EAL f is_integrable_on M by A1, MESFUNC6:def 9;
then consider E being Element of S such that
A4: E = (dom (R_EAL f)) /\ (dom (R_EAL (- g))) and
A5: Integral M,((R_EAL f) + (R_EAL (- g))) = (Integral M,((R_EAL f) | E)) + (Integral M,((R_EAL (- g)) | E)) by A3, MESFUNC5:115;
take E ; :: thesis: ( E = (dom f) /\ (dom g) & Integral M,(f - g) = (Integral M,(f | E)) + (Integral M,((- g) | E)) )
dom (R_EAL (- g)) = dom (- (R_EAL g)) by MESFUNC6:28;
hence ( E = (dom f) /\ (dom g) & Integral M,(f - g) = (Integral M,(f | E)) + (Integral M,((- g) | E)) ) by A4, A5, MESFUNC1:def 7, MESFUNC6:23; :: thesis: verum