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 st ex A being Element of S st
( A = dom f & f is_measurable_on A ) & f is nonnegative holds
0 <= Integral M,f
let S be SigmaField of X; :: thesis: for M being sigma_Measure of S
for f being PartFunc of X,REAL st ex A being Element of S st
( A = dom f & f is_measurable_on A ) & f is nonnegative holds
0 <= Integral M,f
let M be sigma_Measure of S; :: thesis: for f being PartFunc of X,REAL st ex A being Element of S st
( A = dom f & f is_measurable_on A ) & f is nonnegative holds
0 <= Integral M,f
let f be PartFunc of X,REAL ; :: thesis: ( ex A being Element of S st
( A = dom f & f is_measurable_on A ) & f is nonnegative implies 0 <= Integral M,f )
assume that
A1: ex A being Element of S st
( A = dom f & f is_measurable_on A ) and
A2: f is nonnegative ; :: thesis: 0 <= Integral M,f
consider A being Element of S such that
A3: A = dom f and
A4: f is_measurable_on A by A1;
R_EAL f is_measurable_on A by A4, Def6;
hence 0 <= Integral M,f by A2, A3, MESFUNC5:96; :: thesis: verum