let X be non empty set ; :: thesis: 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; :: thesis: 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; :: thesis: 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; :: thesis: ( 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 ; :: thesis: 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; :: thesis: verum