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

theorem Th4:
  for M be sigma_Measure of S, F be SetSequence of S st
  M.(Union F) < +infty holds
  ex G be sequence of S st G= superior_setsequence F &
  M.(lim_sup F) = inf rng(M*G)
proof
  let M be sigma_Measure of S, F be SetSequence of S;
  rng superior_setsequence F c= S;
  then reconsider G= superior_setsequence F as sequence of S by FUNCT_2:6;
  reconsider F1 = F, G1 = G as SetSequence of X;
A1: for n being Nat holds G.(n+1) c= G.n by NAT_1:12,PROB_1:def 4;
  G.0 = union {F.k where k: 0 <= k} by SETLIM_1:def 3; then
A2: G.0 = union rng F by SETLIM_1:5;
  consider f being SetSequence of X such that
A3: lim_sup F1 = meet f and
A4: for n being Nat holds f.n = Union (F1^\n) by KURATO_0:def 2;
  now
    let n be Element of NAT;
    f.n = Union (F1^\n) by A4;
    hence f.n = G1.n by Th2;
  end; then
A5: f = G1 by FUNCT_2:63;
  assume M.(Union F) < +infty;
  then M.(meet rng G)= inf rng(M*G) by A1,A2,MEASURE3:12;
  hence thesis by A3,A5;
end;
