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 ) & f is nonnegative holds
0 <= Integral M,(f | A)
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 ) & f is nonnegative holds
0 <= Integral M,(f | A)
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 ) & f is nonnegative holds
0 <= Integral M,(f | A)
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 ) & f is nonnegative holds
0 <= Integral M,(f | A)
let A be Element of S; :: thesis: ( ex E being Element of S st
( E = dom f & f is_measurable_on E ) & f is nonnegative implies 0 <= Integral M,(f | A) )
assume A1: ( ex E being Element of S st
( E = dom f & f is_measurable_on E ) & f is nonnegative ) ; :: thesis: 0 <= Integral M,(f | A)
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 0 <= Integral M,(f | A) by A1, MESFUNC5:98; :: thesis: verum