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 that
A1: ex E being Element of S st
( E = dom f & f is_measurable_on E ) and
A2: f is nonnegative ; :: thesis: 0 <= Integral (M,(f | A))
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 0 <= Integral (M,(f | A)) by A2, A3, MESFUNC5:92; :: thesis: verum