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_measurable_on A iff for r being real number holds A /\ (great_eq_dom f,(R_EAL 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_measurable_on A iff for r being real number holds A /\ (great_eq_dom f,(R_EAL 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_measurable_on A iff for r being real number holds A /\ (great_eq_dom f,(R_EAL r)) in S )
let A be Element of S; :: thesis: ( A c= dom f implies ( f is_measurable_on A iff for r being real number holds A /\ (great_eq_dom f,(R_EAL r)) in S ) )
assume A1: A c= dom f ; :: thesis: ( f is_measurable_on A iff for r being real number holds A /\ (great_eq_dom f,(R_EAL r)) in S )
A2: ( f is_measurable_on A implies for r being real number holds A /\ (great_eq_dom f,(R_EAL r)) in S )

proof

assume A3: f is_measurable_on A ; :: thesis: for r being real number holds A /\ (great_eq_dom f,(R_EAL r)) in S
for r being real number holds A /\ (great_eq_dom f,(R_EAL r)) in S

proof

let r be real number ; :: thesis: A /\ (great_eq_dom f,(R_EAL r)) in S
A4: A /\ (less_dom f,(R_EAL r)) in S by A3, Def17;
A /\ (great_eq_dom f,(R_EAL r)) = A \ (A /\ (less_dom f,(R_EAL r))) by A1, Th18;
hence A /\ (great_eq_dom f,(R_EAL r)) in S by A4, MEASURE1:20; :: thesis: verum

end;

hence for r being real number holds A /\ (great_eq_dom f,(R_EAL r)) in S ; :: thesis: verum

end;

( ( for r being real number holds A /\ (great_eq_dom f,(R_EAL r)) in S ) implies f is_measurable_on A )

proof

assume A5: for r being real number holds A /\ (great_eq_dom f,(R_EAL r)) in S ; :: thesis: f is_measurable_on A
for r being real number holds A /\ (less_dom f,(R_EAL r)) in S

proof

let r be real number ; :: thesis: A /\ (less_dom f,(R_EAL r)) in S
A6: A /\ (great_eq_dom f,(R_EAL r)) in S by A5;
A /\ (less_dom f,(R_EAL r)) = A \ (A /\ (great_eq_dom f,(R_EAL r))) by A1, Th21;
hence A /\ (less_dom f,(R_EAL r)) in S by A6, MEASURE1:20; :: thesis: verum

end;

hence f is_measurable_on A by Def17; :: thesis: verum

end;

hence ( f is_measurable_on A iff for r being real number holds A /\ (great_eq_dom f,(R_EAL r)) in S ) by A2; :: thesis: verum