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

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