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 being Element of S st ex E being Element of S st
( E = dom f & f is_measurable_on E ) & M . A = 0 holds
Integral M,(f | A) = 0
let S be SigmaField of X; :: thesis: for M being sigma_Measure of S
for f being PartFunc of X,REAL
for 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
Integral M,(f | A) = 0
let M be sigma_Measure of S; :: thesis: for f being PartFunc of X,REAL
for 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
Integral M,(f | A) = 0
let f be PartFunc of X,REAL ; :: thesis: for 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
Integral M,(f | A) = 0
let 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 Integral M,(f | A) = 0 )
assume A1: ( ex E being Element of S st
( E = dom f & f is_measurable_on E ) & M . A = 0 ) ; :: thesis: Integral M,(f | A) = 0
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, Def6;
hence Integral M,(f | A) = 0 by A1, MESFUNC5:100; :: thesis: verum