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_measurable_on A & g is_measurable_on A holds
f + g is_measurable_on A
let S be SigmaField of X; :: thesis: 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 holds
f + g is_measurable_on A
let f, g be PartFunc of X,COMPLEX; :: thesis: for A being Element of S st f is_measurable_on A & g is_measurable_on A holds
f + g is_measurable_on A
let A be Element of S; :: thesis: ( f is_measurable_on A & g is_measurable_on A implies f + g is_measurable_on A )
assume that
A1: f is_measurable_on A and
A2: g is_measurable_on A ; :: thesis: f + g is_measurable_on A
A3: Im g is_measurable_on A by A2, Def3;
Im f is_measurable_on A by A1, Def3;
then (Im f) + (Im g) is_measurable_on A by A3, MESFUNC6:26;
then A4: Im (f + g) is_measurable_on A by Th5;
A5: Re g is_measurable_on A by A2, Def3;
Re f is_measurable_on A by A1, Def3;
then (Re f) + (Re g) is_measurable_on A by A5, MESFUNC6:26;
then Re (f + g) is_measurable_on A by Th5;
hence f + g is_measurable_on A by A4, Def3; :: thesis: verum