let X be non empty set ; :: thesis: for S being SigmaField of X
for f being PartFunc of X,ExtREAL
for A being Element of S holds
( f is A -measurable iff for r being Real holds A /\ (less_eq_dom (f,r)) in S )
let S be SigmaField of X; :: thesis: for f being PartFunc of X,ExtREAL
for A being Element of S holds
( f is A -measurable iff for r being Real holds A /\ (less_eq_dom (f,r)) in S )
let f be PartFunc of X,ExtREAL; :: thesis: for A being Element of S holds
( f is A -measurable iff for r being Real holds A /\ (less_eq_dom (f,r)) in S )
let A be Element of S; :: thesis: ( f is A -measurable iff for r being Real holds A /\ (less_eq_dom (f,r)) in S )
A1: ( f is A -measurable implies for r being Real holds A /\ (less_eq_dom (f,r)) in S )

proof

assume A2: f is A -measurable ; :: thesis: for r being Real holds A /\ (less_eq_dom (f,r)) in S
for r being Real holds A /\ (less_eq_dom (f,r)) in S

proof

let r be Real; :: thesis: A /\ (less_eq_dom (f,r)) in S
defpred S₁[ Element of NAT , set ] means A /\ (less_dom (f,(r + (1 / ($1 + 1))))) = $2;
A3: for n being Element of NAT ex y being Element of S st S₁[n,y]

proof

let n be Element of NAT ; :: thesis: ex y being Element of S st S₁[n,y]
reconsider y = A /\ (less_dom (f,(r + (1 / (n + 1))))) as Element of S by A2;
take y ; :: thesis: S₁[n,y]
thus S₁[n,y] ; :: thesis: verum

end;

consider F being sequence of S such that
A4: for n being Element of NAT holds S₁[n,F . n] from FUNCT_2:sch 3(A3);
A /\ (less_eq_dom (f,r)) = meet (rng F) by A4, Th20;
hence A /\ (less_eq_dom (f,r)) in S ; :: thesis: verum

end;

hence for r being Real holds A /\ (less_eq_dom (f,r)) in S ; :: thesis: verum

end;

( ( for r being Real holds A /\ (less_eq_dom (f,r)) in S ) implies f is A -measurable )

proof

assume A5: for r being Real holds A /\ (less_eq_dom (f,r)) in S ; :: thesis: f is A -measurable
for r being Real holds A /\ (less_dom (f,r)) in S

proof

let r be Real; :: thesis: A /\ (less_dom (f,r)) in S
defpred S₁[ Element of NAT , set ] means A /\ (less_eq_dom (f,(r - (1 / ($1 + 1))))) = $2;
A6: for n being Element of NAT ex y being Element of S st S₁[n,y]

proof

let n be Element of NAT ; :: thesis: ex y being Element of S st S₁[n,y]
reconsider y = A /\ (less_eq_dom (f,(r - (1 / (n + 1))))) as Element of S by A5;
take y ; :: thesis: S₁[n,y]
thus S₁[n,y] ; :: thesis: verum

end;

consider F being sequence of S such that
A7: for n being Element of NAT holds S₁[n,F . n] from FUNCT_2:sch 3(A6);
A /\ (less_dom (f,r)) = union (rng F) by A7, Th21;
hence A /\ (less_dom (f,r)) in S ; :: thesis: verum

end;

hence f is A -measurable ; :: thesis: verum

end;

hence ( f is A -measurable iff for r being Real holds A /\ (less_eq_dom (f,r)) in S ) by A1; :: thesis: verum