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 holds

( max+ f is nonnegative & max- f is nonnegative & |.f.| is nonnegative )

let S be SigmaField of X; :: thesis: for M being sigma_Measure of S

for f being PartFunc of X,ExtREAL holds

( max+ f is nonnegative & max- f is nonnegative & |.f.| is nonnegative )

let M be sigma_Measure of S; :: thesis: for f being PartFunc of X,ExtREAL holds

( max+ f is nonnegative & max- f is nonnegative & |.f.| is nonnegative )

let f be PartFunc of X,ExtREAL; :: thesis: ( max+ f is nonnegative & max- f is nonnegative & |.f.| is nonnegative )

A1: for x being object st x in dom (max- f) holds

0 <= (max- f) . x by MESFUNC2:13;

for x being object st x in dom (max+ f) holds

0 <= (max+ f) . x by MESFUNC2:12;

hence ( max+ f is nonnegative & max- f is nonnegative ) by A1, SUPINF_2:52; :: thesis: |.f.| is nonnegative

for M being sigma_Measure of S

for f being PartFunc of X,ExtREAL holds

( max+ f is nonnegative & max- f is nonnegative & |.f.| is nonnegative )

let S be SigmaField of X; :: thesis: for M being sigma_Measure of S

for f being PartFunc of X,ExtREAL holds

( max+ f is nonnegative & max- f is nonnegative & |.f.| is nonnegative )

let M be sigma_Measure of S; :: thesis: for f being PartFunc of X,ExtREAL holds

( max+ f is nonnegative & max- f is nonnegative & |.f.| is nonnegative )

let f be PartFunc of X,ExtREAL; :: thesis: ( max+ f is nonnegative & max- f is nonnegative & |.f.| is nonnegative )

A1: for x being object st x in dom (max- f) holds

0 <= (max- f) . x by MESFUNC2:13;

for x being object st x in dom (max+ f) holds

0 <= (max+ f) . x by MESFUNC2:12;

hence ( max+ f is nonnegative & max- f is nonnegative ) by A1, SUPINF_2:52; :: thesis: |.f.| is nonnegative

now :: thesis: for x being object st x in dom |.f.| holds

0 <= |.f.| . x

hence
|.f.| is nonnegative
by SUPINF_2:52; :: thesis: verum0 <= |.f.| . x

let x be object ; :: thesis: ( x in dom |.f.| implies 0 <= |.f.| . x )

assume x in dom |.f.| ; :: thesis: 0 <= |.f.| . x

then |.f.| . x = |.(f . x).| by MESFUNC1:def 10;

hence 0 <= |.f.| . x by EXTREAL1:14; :: thesis: verum

end;assume x in dom |.f.| ; :: thesis: 0 <= |.f.| . x

then |.f.| . x = |.(f . x).| by MESFUNC1:def 10;

hence 0 <= |.f.| . x by EXTREAL1:14; :: thesis: verum