reserve X for non empty set,
  Y for set,
  S for SigmaField of X,
  M for sigma_Measure of S,
  f,g for PartFunc of X,COMPLEX,
  r for Real,
  k for Real,
  n for Nat,
  E for Element of S;

theorem
  for f be with_the_same_dom Functional_Sequence of X,REAL, 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,REAL, E be Element of S;
  assume that
A1: dom(f.0) = E and
A2: for n be Nat holds f.n is E-measurable;
  for n being Nat holds (R_EAL f).n is E-measurable 
  proof
    let n be Nat; 
    f.n is E-measurable by A2;
    hence thesis by Th7;
  end;
  hence thesis by A1,MESFUNC8:24;
end;
