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,ExtREAL st ex A being Element of S st
( A = dom f & f is A -measurable ) & 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,ExtREAL st ex A being Element of S st
( A = dom f & f is A -measurable ) & f is nonnegative holds
0 <= Integral (M,f)
let M be sigma_Measure of S; :: thesis: for f being PartFunc of X,ExtREAL st ex A being Element of S st
( A = dom f & f is A -measurable ) & f is nonnegative holds
0 <= Integral (M,f)
let f be PartFunc of X,ExtREAL; :: thesis: ( ex A being Element of S st
( A = dom f & f is A -measurable ) & f is nonnegative implies 0 <= Integral (M,f) )
assume that
A1: ex A being Element of S st
( A = dom f & f is A -measurable ) and
A2: f is nonnegative ; :: thesis: 0 <= Integral (M,f)
0 <= integral+ (M,f) by A1, A2, Th79;
hence 0 <= Integral (M,f) by A1, A2, Th88; :: thesis: verum