So let be complete -finite measure spaces. Then consider , the completion of .

This is the basic set up. I have to show that if is -measurable and almost everywhere, then are integrable and almost everywhere.

Note: for fixed x,y.

I was told that for this part that I need to use the fact that are both complete but I don't see how.

I am starting out by assuming that (characteristic function). Then right? I assume that it is at this point I need to make use of completeness some how.