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 )
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 )
let M be sigma_Measure of S; :: thesis: for f being PartFunc of X,ExtREAL holds
( max+ f is nonnegative & max- f is nonnegative )
let f be PartFunc of X,ExtREAL ; :: thesis: ( max+ f is nonnegative & max- f is nonnegative )
( ( for x being set st x in dom (max+ f) holds
0 <= (max+ f) . x ) & ( for x being set st x in dom (max- f) holds
0 <= (max- f) . x ) ) by MESFUNC2:14, MESFUNC2:15;
hence ( max+ f is nonnegative & max- f is nonnegative ) by SUPINF_2:71; :: thesis: verum