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) /\ (dom g) & 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) /\ (dom g) & 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) /\ (dom g) & 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) /\ (dom g) & 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) /\ (dom g) & 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) /\ (dom g) and
A2: f is_measurable_on A and
A3: g is_measurable_on A and
A4: f is without-infty and
A5: g is without-infty ; :: thesis: (max- (f + g)) + (max+ f) is_measurable_on A
A6: dom (f + g) = (dom f) /\ (dom g) by A4, A5, Th22;
f + g is_measurable_on A by A2, A3, A4, A5, Th37;
then A7: max- (f + g) is_measurable_on A by A1, A6, MESFUNC2:28;
A8: max- (f + g) is nonnegative by Lm1;
A9: max+ f is nonnegative by Lm1;
max+ f is_measurable_on A by A2, MESFUNC2:27;
hence (max- (f + g)) + (max+ f) is_measurable_on A by A7, A8, A9, Th37; :: thesis: verum