let Omega be non empty set ; :: thesis:
let Sigma be SigmaField of Omega; :: thesis:
let s be Real; :: thesis:
let X be Function of Omega,REAL; :: thesis:
assume A0: X = Omega --> s ; :: thesis:
X is Sigma, Borel_Sets -random_variable-like

proof

let x be set ; :: according to FINANCE1:def 5 :: thesis:

per cases ;

suppose A1: s in x ; :: thesis:

for q being object holds
( q in { w where w is Element of Omega : X . w is Element of x } iff q in Omega )

proof

let q be object ; :: thesis:

hereby :: thesis:

assume q in { w where w is Element of Omega : X . w is Element of x } ; :: thesis:
then ex w being Element of Omega st
( w = q & X . w is Element of x ) ;
hence q in Omega ; :: thesis:

end;

assume q in Omega ; :: thesis:
then reconsider w = q as Element of Omega ;
X . w is Element of x by A1, FUNCOP_1:7, A0;
hence q in { w where w is Element of Omega : X . w is Element of x } ; :: thesis:

end;

then A2: { w where w is Element of Omega : X . w is Element of x } = Omega by TARSKI:2;
not x is empty by A1;
then { w where w is Element of Omega : X . w is Element of x } = X " x by Lm1;
then X " x = Omega by A2;
then X " x is Element of Sigma by PROB_1:23;
hence ( x in Borel_Sets implies X " x in Sigma ) ; :: thesis:

end;

suppose A3: not s in x ; :: thesis:

for q being object holds
( q in X " x iff q in {} )

proof

let q be object ; :: thesis:

now :: thesis:

assume q in X " x ; :: thesis:
then ( q in dom X & X . q in x ) by FUNCT_1:def 7;
hence q in {} by A3, FUNCOP_1:7, A0; :: thesis:

end;

hence ( q in X " x iff q in {} ) ; :: thesis:

end;

then X " x = {} by TARSKI:2;
then X " x is Element of Sigma by PROB_1:22;
hence ( x in Borel_Sets implies X " x in Sigma ) ; :: thesis:

end;

end;

end;

hence X is Sigma, Borel_Sets -random_variable-like ; :: thesis: