let X be non empty set ; :: thesis: for S being SigmaField of X
for f being PartFunc of X,REAL
for A, B being Element of S st dom f = A holds
( f is_measurable_on B iff f is_measurable_on A /\ B )
let S be SigmaField of X; :: thesis: for f being PartFunc of X,REAL
for A, B being Element of S st dom f = A holds
( f is_measurable_on B iff f is_measurable_on A /\ B )
let f be PartFunc of X,REAL ; :: thesis: for A, B being Element of S st dom f = A holds
( f is_measurable_on B iff f is_measurable_on A /\ B )
let A, B be Element of S; :: thesis: ( dom f = A implies ( f is_measurable_on B iff f is_measurable_on A /\ B ) )
assume A1: dom f = A ; :: thesis: ( f is_measurable_on B iff f is_measurable_on A /\ B )

A2: now

let r be real number ; :: thesis: A /\ (less_dom f,r) = less_dom f,r

now

let x be set ; :: thesis: ( x in A /\ (less_dom f,r) iff x in less_dom f,r )
( x in A /\ (less_dom f,r) iff ( x in A & x in less_dom f,r ) ) by XBOOLE_0:def 4;
hence ( x in A /\ (less_dom f,r) iff x in less_dom f,r ) by A1, MESFUNC1:def 12; :: thesis: verum

end;

then ( A /\ (less_dom f,r) c= less_dom f,r & less_dom f,r c= A /\ (less_dom f,r) ) by TARSKI:def 3;
hence A /\ (less_dom f,r) = less_dom f,r by XBOOLE_0:def 10; :: thesis: verum

end;

hereby :: thesis: ( f is_measurable_on A /\ B implies f is_measurable_on B )

assume A3: f is_measurable_on B ; :: thesis: f is_measurable_on A /\ B

now

let r be real number ; :: thesis: (A /\ B) /\ (less_dom f,r) in S
(A /\ B) /\ (less_dom f,r) = B /\ (A /\ (less_dom f,r)) by XBOOLE_1:16
.= B /\ (less_dom f,r) by A2 ;
hence (A /\ B) /\ (less_dom f,r) in S by A3, Th12; :: thesis: verum

end;

hence f is_measurable_on A /\ B by Th12; :: thesis: verum

end;

assume A4: f is_measurable_on A /\ B ; :: thesis: f is_measurable_on B

now

let r be real number ; :: thesis: B /\ (less_dom f,r) in S
(A /\ B) /\ (less_dom f,r) = B /\ (A /\ (less_dom f,r)) by XBOOLE_1:16
.= B /\ (less_dom f,r) by A2 ;
hence B /\ (less_dom f,r) in S by A4, Th12; :: thesis: verum

end;

hence f is_measurable_on B by Th12; :: thesis: verum