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 st A c= dom f holds
( f is A -measurable iff for r being Real holds A /\ (great_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 st A c= dom f holds
( f is A -measurable iff for r being Real holds A /\ (great_dom (f,r)) in S )
let f be PartFunc of X,ExtREAL; :: thesis: for A being Element of S st A c= dom f holds
( f is A -measurable iff for r being Real holds A /\ (great_dom (f,r)) in S )
let A be Element of S; :: thesis: ( A c= dom f implies ( f is A -measurable iff for r being Real holds A /\ (great_dom (f,r)) in S ) )
assume A1: A c= dom f ; :: thesis: ( f is A -measurable iff for r being Real holds A /\ (great_dom (f,r)) in S )
A2: ( f is A -measurable implies for r being Real holds A /\ (great_dom (f,r)) in S )

proof

assume A3: f is A -measurable ; :: thesis: for r being Real holds A /\ (great_dom (f,r)) in S
for r being Real holds A /\ (great_dom (f,r)) in S

proof

let r be Real; :: thesis: A /\ (great_dom (f,r)) in S
( A /\ (less_eq_dom (f,r)) in S & A /\ (great_dom (f,r)) = A \ (A /\ (less_eq_dom (f,r))) ) by A1, A3, Th15, Th28;
hence A /\ (great_dom (f,r)) in S by MEASURE1:6; :: thesis: verum

end;

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

end;

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

proof

assume A4: for r being Real holds A /\ (great_dom (f,r)) in S ; :: thesis: f is A -measurable
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
( A /\ (great_dom (f,r)) in S & A /\ (less_eq_dom (f,r)) = A \ (A /\ (great_dom (f,r))) ) by A1, A4, Th16;
hence A /\ (less_eq_dom (f,r)) in S by MEASURE1:6; :: thesis: verum

end;

hence f is A -measurable by Th28; :: thesis: verum

end;

hence ( f is A -measurable iff for r being Real holds A /\ (great_dom (f,r)) in S ) by A2; :: thesis: verum