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

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