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?