If where , the -Algebra of X, then a function f on X is measurable iff f is measurable on A and on B.

Proof so far.

Assume that , where are the set of all -Algebra of X and Y, respectively.

First, assume that f is measurable on X, that is, I have

Consider the function , I need to show that , where is the a -Algebra of A.

I know that has to be true, but I'm just having a hard time trying to write up the sigma algebra of A in X, any hints?

Thank you.