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