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

theorem Th23:
  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) holds lim_sup f is E-measurable
proof
  let f be with_the_same_dom Functional_Sequence of X,ExtREAL, E be Element of
  S;
  assume that
A1: dom (f.0) = E and
A2: for n be Nat holds f.n is E-measurable;
A3: now
    let r be Real;
    deffunc G(Element of NAT) = E /\ great_eq_dom((superior_realsequence f).$1
    ,r);
    consider F being sequence of bool X such that
A4: for x being Element of NAT holds F.x = G(x) from FUNCT_2:sch 4;
    now
      let i be Nat;
A5:   dom((superior_realsequence f).i) = dom(f.0) by Def6;
      i in NAT by ORDINAL1:def 12;
      then
A6:   F.i = E /\ great_eq_dom((superior_realsequence f).i,r) by A4;
      (superior_realsequence f).i is E-measurable by A1,A2,Th19;
      hence F.i in S by A1,A6,A5,MESFUNC1:27;
    end;
    then
A7: rng F c= S by NAT_1:52;
A8: for x being Nat holds
      F.x = E /\ great_eq_dom((superior_realsequence f).x,r)
    proof
      let x be Nat;
      reconsider x9=x as Element of NAT by ORDINAL1:def 12;
      F.x9 = E /\ great_eq_dom((superior_realsequence f).x9,r) by A4;
      hence thesis;
    end;
    reconsider F as SetSequence of S by A7,RELAT_1:def 19;
    rng F c= S;
    then F is sequence of S by FUNCT_2:6;
    then
A9: rng F is N_Sub_set_fam of X by MEASURE1:23;
A10: rng F is N_Measure_fam of S by A9,MEASURE2:def 1;
    meet F = E /\ great_eq_dom(lim_sup f,r) by A1,A8,Th21;
    hence E /\ great_eq_dom(lim_sup f,r) in S by A10,MEASURE2:2;
  end;
  dom (lim_sup f) = dom(f.0) by Def8;
  hence thesis by A1,A3,MESFUNC1:27;
end;
