let X be non empty set ; :: thesis: for S being SigmaField of X
for f being PartFunc of X,COMPLEX
for B, A being Element of S st f is_measurable_on B & A = (dom f) /\ B holds
f | B is_measurable_on A
let S be SigmaField of X; :: thesis: for f being PartFunc of X,COMPLEX
for B, A being Element of S st f is_measurable_on B & A = (dom f) /\ B holds
f | B is_measurable_on A
let f be PartFunc of X,COMPLEX ; :: thesis: for B, A being Element of S st f is_measurable_on B & A = (dom f) /\ B holds
f | B is_measurable_on A
let B, A be Element of S; :: thesis: ( f is_measurable_on B & A = (dom f) /\ B implies f | B is_measurable_on A )
assume A1: ( f is_measurable_on B & A = (dom f) /\ B ) ; :: thesis: f | B is_measurable_on A
then A2: ( Re f is_measurable_on B & Im f is_measurable_on B ) by Def3;
( A = (dom (Re f)) /\ B & A = (dom (Im f)) /\ B ) by A1, Def1, Def2;
then ( (Re f) | B is_measurable_on A & (Im f) | B is_measurable_on A ) by A2, MESFUNC6:76;
then ( Re (f | B) is_measurable_on A & Im (f | B) is_measurable_on A ) by COM91;
hence f | B is_measurable_on A by Def3; :: thesis: verum