 reserve Omega, Omega2 for non empty set;
 reserve Sigma, F for SigmaField of Omega;
 reserve Sigma2, F2 for SigmaField of Omega2;

theorem Th15:
for f being Function of Omega,REAL st
    f is (Sigma,Borel_Sets)-random_variable-like holds
          f is ([#]Sigma)-measurable &
          f is Real-Valued-Random-Variable of Sigma
proof
 let f be Function of Omega,REAL;
 assume A1: f is (Sigma,Borel_Sets)-random_variable-like;
A2: for r being Element of REAL holds Omega /\ less_dom(f,r) in Sigma
 proof
  let r be Element of REAL;
  less_dom(f,r) = {w where w is Element of Omega: f.w <r} by A1,Th9;
  then less_dom(f,r) is Element of Sigma by A1,Th9; then
  less_dom(f,r) in Sigma;
  hence thesis by XBOOLE_1:28;
 end;
for r being Real holds [#]Sigma /\ less_dom(f,r) in Sigma
proof
 let r be Real;
 reconsider r as Element of REAL by XREAL_0:def 1;
 [#]Sigma /\ less_dom(f,r) in Sigma by A2;
 hence thesis;
end;
hence thesis by MESFUNC6:12;
end;
