
theorem Th30:
for X be non empty set, S be SigmaField of X, M be sigma_Measure of S,
 E1 be Element of S, E2 be Element of COM(S,M), f be PartFunc of X,ExtREAL
 st E1 = E2 & f is E1-measurable holds f is E2-measurable
proof
    let X be non empty set, S be SigmaField of X, M be sigma_Measure of S,
    E1 be Element of S, E2 be Element of COM(S,M), f be PartFunc of X,ExtREAL;
    assume that
A1:  E1 = E2 and
A2:  f is E1-measurable;
    for r be Real holds E2 /\ less_dom(f,r) in COM(S,M)
    proof
     let r be Real;
     E1 /\ less_dom(f,r) in S by A2,MESFUNC1:def 16; then
     E2 /\ less_dom(f,r) is Element of COM(S,M) by A1,Th27;
     hence E2 /\ less_dom(f,r) in COM(S,M);
    end;
    hence f is E2-measurable by MESFUNC1:def 16;
end;
