let X be non empty set ; :: thesis: for S being SigmaField of X
for M being sigma_Measure of S
for A being Element of S
for f, g being PartFunc of X,ExtREAL st A c= dom f & f is_measurable_on A & g is_measurable_on A & f is without-infty & g is without-infty holds
(max+ (f + g)) + (max- f) is_measurable_on A
let S be SigmaField of X; :: thesis: for M being sigma_Measure of S
for A being Element of S
for f, g being PartFunc of X,ExtREAL st A c= dom f & f is_measurable_on A & g is_measurable_on A & f is without-infty & g is without-infty holds
(max+ (f + g)) + (max- f) is_measurable_on A
let M be sigma_Measure of S; :: thesis: for A being Element of S
for f, g being PartFunc of X,ExtREAL st A c= dom f & f is_measurable_on A & g is_measurable_on A & f is without-infty & g is without-infty holds
(max+ (f + g)) + (max- f) is_measurable_on A
let A be Element of S; :: thesis: for f, g being PartFunc of X,ExtREAL st A c= dom f & f is_measurable_on A & g is_measurable_on A & f is without-infty & g is without-infty holds
(max+ (f + g)) + (max- f) is_measurable_on A
let f, g be PartFunc of X,ExtREAL ; :: thesis: ( A c= dom f & f is_measurable_on A & g is_measurable_on A & f is without-infty & g is without-infty implies (max+ (f + g)) + (max- f) is_measurable_on A )
assume that
A1: A c= dom f and
A2: ( f is_measurable_on A & g is_measurable_on A ) and
A3: ( f is without-infty & g is without-infty ) ; :: thesis: (max+ (f + g)) + (max- f) is_measurable_on A
f + g is_measurable_on A by A2, A3, Th37;
then A4: max+ (f + g) is_measurable_on A by MESFUNC2:27;
A5: max- f is_measurable_on A by A1, A2, MESFUNC2:28;
( max+ (f + g) is nonnegative & max- f is nonnegative ) by Lm1;
hence (max+ (f + g)) + (max- f) is_measurable_on A by A4, A5, Th37; :: thesis: verum