I am asked to prove that, if

and

, prove that

I doubt that I want to do this by brute force by providing some increasing sequence of upper functions that approach

, though maybe I do since I know that

and likewise for

.

In any case, the alternative seems to be to use to the Dominated Convergence Theorem, in which case I want to show that

converges almost everywhere (presumably to

), and that there is a non-negative function which is integrable and dominates

. But the first part seems like it might be false, since I don't know that the interval over which these functions are integrated is bounded.

Maybe I want to try an epsilon-delta argument?