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
( A /\ (less_dom (f,(R_EAL r))) in S & A /\ (great_eq_dom (f,(R_EAL r))) = A \ (A /\ (less_dom (f,(R_EAL r)))) ) by A1, A3, Def17, Th18;
hence A /\ (great_eq_dom (f,(R_EAL r))) in S by MEASURE1:6; :: 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 A4: 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
( A /\ (great_eq_dom (f,(R_EAL r))) in S & A /\ (less_dom (f,(R_EAL r))) = A \ (A /\ (great_eq_dom (f,(R_EAL r)))) ) by A1, A4, Th21;
hence A /\ (less_dom (f,(R_EAL r))) in S by MEASURE1:6; :: 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