let X be non empty set ; for S being SigmaField of X
for f, g being PartFunc of X,COMPLEX
for A being Element of S st f is A -measurable & g is A -measurable & A c= dom g holds
f - g is A -measurable
let S be SigmaField of X; for f, g being PartFunc of X,COMPLEX
for A being Element of S st f is A -measurable & g is A -measurable & A c= dom g holds
f - g is A -measurable
let f, g be PartFunc of X,COMPLEX; for A being Element of S st f is A -measurable & g is A -measurable & A c= dom g holds
f - g is A -measurable
let A be Element of S; ( f is A -measurable & g is A -measurable & A c= dom g implies f - g is A -measurable )
assume that
A1:
f is A -measurable
and
A2:
g is A -measurable
and
A3:
A c= dom g
; f - g is A -measurable
A4:
Im g is A -measurable
by A2;
A5:
A c= dom (Re g)
by A3, COMSEQ_3:def 3;
A6:
Re g is A -measurable
by A2;
A7:
A c= dom (Im g)
by A3, COMSEQ_3:def 4;
Im f is A -measurable
by A1;
then
(Im f) - (Im g) is A -measurable
by A4, A7, MESFUNC6:29;
then A8:
Im (f - g) is A -measurable
by Th6;
Re f is A -measurable
by A1;
then
(Re f) - (Re g) is A -measurable
by A6, A5, MESFUNC6:29;
then
Re (f - g) is A -measurable
by Th6;
hence
f - g is A -measurable
by A8; verum