What about uniform convergence? It seems that you are not given enough information though.
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?
Notice then that we only require to coverge weakly to and to converge in norm. The proof also shows that th result holds if and with and .