reserve Omega for non empty set;
reserve r for Real;
reserve Sigma for SigmaField of Omega;
reserve P for Probability of Sigma;
reserve E for finite non empty set;
reserve f,g for Real-Valued-Random-Variable of Sigma;

theorem Th30:
  for Omega be non empty finite set, P be Probability of
  Trivial-SigmaField (Omega), X be Real-Valued-Random-Variable of
  Trivial-SigmaField (Omega) holds X is_integrable_on P
proof
  let Omega be non empty finite set, P be Probability of Trivial-SigmaField (
  Omega), X be Real-Valued-Random-Variable of Trivial-SigmaField (Omega);
  set M= P2M(P);
A1: jj in REAL by XREAL_0:def 1;
  dom X = Omega by FUNCT_2:def 1;
  then M.(dom X) = jj by PROB_1:def 8;
  then M.(dom X) < +infty by XXREAL_0:9,A1;
  then X is_integrable_on M by Lm5,Th6;
  hence thesis;
end;
