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 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 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 holds
f + g is A -measurable
let A be Element of S; :: thesis: ( f is A -measurable & g is A -measurable implies f + g is A -measurable )
assume that
A1: f is A -measurable and
A2: g is A -measurable ; :: thesis: f + g is A -measurable
A3: Im g is A -measurable by A2;
Im f is A -measurable by A1;
then (Im f) + (Im g) is A -measurable by A3, MESFUNC6:26;
then A4: Im (f + g) is A -measurable by Th5;
A5: Re g is A -measurable by A2;
Re f is A -measurable by A1;
then (Re f) + (Re g) is A -measurable by A5, MESFUNC6:26;
then Re (f + g) is A -measurable by Th5;
hence f + g is A -measurable by A4; :: thesis: verum