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

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