let Omega be non empty set ; :: thesis: for Sigma being SigmaField of Omega
for s being Real holds Omega --> s is_random_variable_on Sigma, Borel_Sets
let Sigma be SigmaField of Omega; :: thesis: for s being Real holds Omega --> s is_random_variable_on Sigma, Borel_Sets
let s be Real; :: thesis: Omega --> s is_random_variable_on Sigma, Borel_Sets
set X = Omega --> s;
Omega --> s is_random_variable_on Sigma, Borel_Sets

proof

let x be set ; :: according to FINANCE1:def 5 :: thesis: ( x in Borel_Sets implies (Omega --> s) " x in Sigma )

per cases ( s in x or not s in x ) ;

suppose A1: s in x ; :: thesis: ( x in Borel_Sets implies (Omega --> s) " x in Sigma )

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

proof

let q be object ; :: thesis: ( q in { w where w is Element of Omega : (Omega --> s) . w is Element of x } iff q in Omega )

hereby :: thesis: ( q in Omega implies q in { w where w is Element of Omega : (Omega --> s) . w is Element of x } )

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

end;

assume q in Omega ; :: thesis: q in { w where w is Element of Omega : (Omega --> s) . w is Element of x }
then reconsider w = q as Element of Omega ;
(Omega --> s) . w is Element of x by A1, FUNCOP_1:7;
hence q in { w where w is Element of Omega : (Omega --> s) . w is Element of x } ; :: thesis: verum

end;

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

end;

suppose A3: not s in x ; :: thesis: ( x in Borel_Sets implies (Omega --> s) " x in Sigma )

for q being object holds
( q in (Omega --> s) " x iff q in {} )

proof

let q be object ; :: thesis: ( q in (Omega --> s) " x iff q in {} )

now :: thesis: ( q in (Omega --> s) " x implies q in {} )

assume q in (Omega --> s) " x ; :: thesis: q in {}
then ( q in dom (Omega --> s) & (Omega --> s) . q in x ) by FUNCT_1:def 7;
hence q in {} by A3, FUNCOP_1:7; :: thesis: verum

end;

hence ( q in (Omega --> s) " x iff q in {} ) ; :: thesis: verum

end;

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

end;

end;

end;

hence Omega --> s is_random_variable_on Sigma, Borel_Sets ; :: thesis: verum