let X be non empty set ; :: thesis: for S being SigmaField of X
for f being PartFunc of X,COMPLEX
for A, B being Element of S st f is B -measurable & A = (dom f) /\ B holds
f | B is A -measurable
let S be SigmaField of X; :: thesis: for f being PartFunc of X,COMPLEX
for A, B being Element of S st f is B -measurable & A = (dom f) /\ B holds
f | B is A -measurable
let f be PartFunc of X,COMPLEX; :: thesis: for A, B being Element of S st f is B -measurable & A = (dom f) /\ B holds
f | B is A -measurable
let A, B be Element of S; :: thesis: ( f is B -measurable & A = (dom f) /\ B implies f | B is A -measurable )
assume that
A1: f is B -measurable and
A2: A = (dom f) /\ B ; :: thesis: f | B is A -measurable
A3: A = (dom (Im f)) /\ B by A2, COMSEQ_3:def 4;
Im f is B -measurable by A1;
then (Im f) | B is A -measurable by A3, MESFUNC6:76;
then A4: Im (f | B) is A -measurable by Th7;
A5: A = (dom (Re f)) /\ B by A2, COMSEQ_3:def 3;
Re f is B -measurable by A1;
then (Re f) | B is A -measurable by A5, MESFUNC6:76;
then Re (f | B) is A -measurable by Th7;
hence f | B is A -measurable by A4; :: thesis: verum