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_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_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_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_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_dom f,(R_EAL r)) in S )
A2: ( f is_measurable_on A implies for r being real number holds A /\ (great_dom f,(R_EAL r)) in S )

proof

assume A3: f is_measurable_on A ; :: thesis: for r being real number holds A /\ (great_dom f,(R_EAL r)) in S
A4: for r being real number holds A /\ (great_dom f,(R_EAL r)) in S

proof

let r be real number ; :: thesis: A /\ (great_dom f,(R_EAL r)) in S
A5: ( A /\ (less_eq_dom f,(R_EAL r)) in S & A /\ (great_dom f,(R_EAL r)) = A \ (A /\ (less_eq_dom f,(R_EAL r))) ) by A1, A3, Th19, Th32;
thus A /\ (great_dom f,(R_EAL r)) in S by A5, MEASURE1:20; :: thesis: verum

end;

thus for r being real number holds A /\ (great_dom f,(R_EAL r)) in S by A4; :: thesis: verum

end;

A6: ( ( for r being real number holds A /\ (great_dom f,(R_EAL r)) in S ) implies f is_measurable_on A )

proof

assume A7: for r being real number holds A /\ (great_dom f,(R_EAL r)) in S ; :: thesis: f is_measurable_on A
A8: for r being real number holds A /\ (less_eq_dom f,(R_EAL r)) in S

proof

let r be real number ; :: thesis: A /\ (less_eq_dom f,(R_EAL r)) in S
A9: ( A /\ (great_dom f,(R_EAL r)) in S & A /\ (less_eq_dom f,(R_EAL r)) = A \ (A /\ (great_dom f,(R_EAL r))) ) by A1, A7, Th20;
thus A /\ (less_eq_dom f,(R_EAL r)) in S by A9, MEASURE1:20; :: thesis: verum

end;

thus f is_measurable_on A by A8, Th32; :: thesis: verum

end;

thus ( f is_measurable_on A iff for r being real number holds A /\ (great_dom f,(R_EAL r)) in S ) by A2, A6; :: thesis: verum