let X be non empty set ; :: thesis: for S being SigmaField of X
for f being PartFunc of X,ExtREAL
for A, B being Element of S st B c= A & f is A -measurable holds
f is B -measurable
let S be SigmaField of X; :: thesis: for f being PartFunc of X,ExtREAL
for A, B being Element of S st B c= A & f is A -measurable holds
f is B -measurable
let f be PartFunc of X,ExtREAL; :: thesis: for A, B being Element of S st B c= A & f is A -measurable holds
f is B -measurable
let A, B be Element of S; :: thesis: ( B c= A & f is A -measurable implies f is B -measurable )
assume that
A1: B c= A and
A2: f is A -measurable ; :: thesis: f is B -measurable
for r being Real holds B /\ (less_dom (f,r)) in S

proof

let r be Real; :: thesis: B /\ (less_dom (f,r)) in S
A3: A /\ (less_dom (f,r)) in S by A2;
B /\ (A /\ (less_dom (f,r))) = (B /\ A) /\ (less_dom (f,r)) by XBOOLE_1:16
.= B /\ (less_dom (f,r)) by A1, XBOOLE_1:28 ;
hence B /\ (less_dom (f,r)) in S by A3, FINSUB_1:def 2; :: thesis: verum

end;

hence f is B -measurable ; :: thesis: verum