let X be non empty set ; :: thesis: for S being SigmaField of X
for f, g being without+infty PartFunc of X,ExtREAL
for A being Element of S st f is A -measurable & g is A -measurable & A c= dom (f + g) holds
f + g is A -measurable
let S be SigmaField of X; :: thesis: for f, g being without+infty PartFunc of X,ExtREAL
for A being Element of S st f is A -measurable & g is A -measurable & A c= dom (f + g) holds
f + g is A -measurable
let f, g be without+infty PartFunc of X,ExtREAL; :: thesis: for A being Element of S st f is A -measurable & g is A -measurable & A c= dom (f + g) holds
f + g is A -measurable
let A be Element of S; :: thesis: ( f is A -measurable & g is A -measurable & A c= dom (f + g) implies f + g is A -measurable )
assume that
A3: f is A -measurable and
A4: g is A -measurable and
A5: A c= dom (f + g) ; :: thesis: f + g is A -measurable
A6: dom (f + g) = (dom f) /\ (dom g) by MESFUNC9:1;
( (dom f) /\ (dom g) c= dom f & (dom f) /\ (dom g) c= dom g ) by XBOOLE_1:17;
then ( A c= dom f & A c= dom g ) by A5, A6;
then ( - f is A -measurable & - g is A -measurable ) by A3, A4, Th59;
then A7: (- f) + (- g) is A -measurable by MESFUNC5:31;
( dom f = dom (- f) & dom g = dom (- g) ) by MESFUNC1:def 7;
then dom ((- f) + (- g)) = (dom f) /\ (dom g) by MESFUNC5:16
.= dom (f + g) by MESFUNC9:1 ;
then A8: - ((- f) + (- g)) is A -measurable by A5, A7, Th59;
(- f) + (- g) = - (f + g) by Th60;
hence f + g is A -measurable by A8, DBLSEQ_3:2; :: thesis: verum