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 that
A1: f is_measurable_on B and
A2: A = (dom f) /\ B ; :: thesis: f | B is_measurable_on A
A3: A = (dom (Im f)) /\ B by A2, COMSEQ_3:def 4;
Im f is_measurable_on B by A1, Def3;
then (Im f) | B is_measurable_on A by A3, MESFUNC6:76;
then A4: Im (f | B) is_measurable_on A by Th7;
A5: A = (dom (Re f)) /\ B by A2, COMSEQ_3:def 3;
Re f is_measurable_on B by A1, Def3;
then (Re f) | B is_measurable_on A by A5, MESFUNC6:76;
then Re (f | B) is_measurable_on A by Th7;
hence f | B is_measurable_on A by A4, Def3; :: thesis: verum