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 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 & 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 & 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 & 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 & 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 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
f + g is A -measurable by A2, A3, A4, A5, Th31;
then A6: max+ (f + g) is A -measurable by MESFUNC2:25;
A7: max- f is nonnegative by Lm1;
A8: max+ (f + g) is nonnegative by Lm1;
max- f is A -measurable by A1, A2, MESFUNC2:26;
hence (max+ (f + g)) + (max- f) is A -measurable by A6, A8, A7, Th31; :: thesis: verum