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 A -measurable & g is A -measurable & f is () & g is () holds
(max- (f + g)) + (max+ f) is A -measurable
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 A -measurable & g is A -measurable & f is () & g is () holds
(max- (f + g)) + (max+ f) is A -measurable
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 A -measurable & g is A -measurable & f is () & g is () holds
(max- (f + g)) + (max+ f) is A -measurable
let A be Element of S; :: thesis: for f, g being PartFunc of X,ExtREAL st A c= (dom f) /\ (dom g) & f is A -measurable & g is A -measurable & f is () & g is () holds
(max- (f + g)) + (max+ f) is A -measurable
let f, g be PartFunc of X,ExtREAL; :: thesis: ( A c= (dom f) /\ (dom g) & f is A -measurable & g is A -measurable & f is () & g is () implies (max- (f + g)) + (max+ f) is A -measurable )
assume that
A1: A c= (dom f) /\ (dom g) and
A2: f is A -measurable and
A3: g is A -measurable and
A4: f is () and
A5: g is () ; :: thesis: (max- (f + g)) + (max+ f) is A -measurable
A6: dom (f + g) = (dom f) /\ (dom g) by A4, A5, Th16;
f + g is A -measurable by A2, A3, A4, A5, Th31;
then A7: max- (f + g) is A -measurable by A1, A6, MESFUNC2:26;
A8: max- (f + g) is nonnegative by Lm1;
A9: max+ f is nonnegative by Lm1;
max+ f is A -measurable by A2, MESFUNC2:25;
hence (max- (f + g)) + (max+ f) is A -measurable by A7, A8, A9, Th31; :: thesis: verum