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

theorem
  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_inf 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_dom((inferior_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;
      i in NAT by ORDINAL1:def 12;
      then
A5:   F.i = E /\ great_dom((inferior_realsequence f).i,r) by A4;
A6:   dom((inferior_realsequence f).i) = E by A1,Def5;
      (inferior_realsequence f).i is E-measurable by A1,A2,Th20;
      hence F.i in S by A5,A6,MESFUNC1:29;
    end;
    then
A7: rng F c= S by NAT_1:52;
A8: for x be Nat holds F.x = E /\ great_dom((
    inferior_realsequence f).x,r)
    proof
      let x be Nat;
      reconsider x9=x as Element of NAT by ORDINAL1:def 12;
      F.x9 = E /\ great_dom((inferior_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;
    union rng F = E /\ great_dom(lim_inf f,r) by A1,A8,Th22;
    hence E /\ great_dom(lim_inf f,r) in S by A10,MEASURE2:2;
  end;
  dom lim_inf f = E by A1,Def7;
  hence thesis by A3,MESFUNC1:29;
end;
