 reserve Omega for non empty set;
 reserve F for SigmaField of Omega;

theorem ZZZ:
  for A being Element of F holds
    chi(A,Omega) is random_variable of F,Borel_Sets
proof
 let A be Element of F;
 set chij=chi(A,Omega);
 chij is Function & rng chij c= REAL & dom chij=Omega
  by FUNCT_3:def 3,VALUED_0:def 3;
 then reconsider chij2 = chij as Function of Omega,REAL by FUNCT_2:2;
 reconsider MyOmega = Omega as Element of F by PROB_1:5;
   chij2 is ([#]F)-measurable by MESFUNC2:29; then
 chij2 is Real-Valued-Random-Variable of F;
 hence thesis by RANDOM_3:7;
end;
