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 ex E being Element of S st
( E = dom f & f is_measurable_on E ) & M . A = 0 holds
f | A is_integrable_on M
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 ex E being Element of S st
( E = dom f & f is_measurable_on E ) & M . A = 0 holds
f | A is_integrable_on M
let M be sigma_Measure of S; :: thesis: for f being PartFunc of X,REAL
for E, 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
f | A is_integrable_on M
let f be PartFunc of X,REAL ; :: thesis: for E, 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
f | A is_integrable_on M
let E, 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 f | A is_integrable_on M )
assume A1: ( ex E being Element of S st
( E = dom f & f is_measurable_on E ) & M . A = 0 ) ; :: thesis: f | A is_integrable_on M
then consider E being Element of S such that
A2: ( E = dom f & f is_measurable_on E ) ;
( E = dom (R_EAL f) & R_EAL f is_measurable_on E ) by A2, MESFUNC6:def 6;
then R_EAL (f | A) is_integrable_on M by A1, Th100a;
hence f | A is_integrable_on M by MESFUNC6:def 9; :: thesis: verum