reserve n,k for Nat,
  X for non empty set,
  S for SigmaField of X;

theorem Th25:
  for f be with_the_same_dom Functional_Sequence of X,ExtREAL,
      E be Element of S st dom(f.0) = E &
  (for n be Nat holds f.n is E-measurable) &
  (for x be Element of X st x in E holds f#x is convergent) holds
  lim f is E-measurable
proof
  let f be with_the_same_dom Functional_Sequence of X,ExtREAL,
      E be Element of S;
  assume
A1: dom (f.0) = E; then
A2: dom lim_sup f = E by Def8;
  assume that
A3: for n be Nat holds f.n is E-measurable and
A4: for x be Element of X st x in E holds f#x is convergent;
A5: dom lim f = E by A1,Def9;
A6: now
    let x be Element of X;
    assume
A7: x in dom lim f;
    then f#x is convergent by A5,A4;
    hence (lim f).x= (lim_sup f).x by A7,Th14;
  end;
  lim_sup f is E-measurable by A1,A3,Th23;
  hence thesis by A5,A2,A6,PARTFUN1:5;
end;
