I think the best you can do is say that is continuous. I've also got . Also, if is the subset of consisting of the equivalence classes of non-negative measurable functions, then .
Isn't there a result along the lines of convergence in implies the existence of a subsequence that converges pointwise almost everywhere? I can't seem to find it.
This works fine, but you would have to use one of the following results:
1) If then for all
2) If , where then and the for fixed , the map is continous (Riesz representation theorem).
Another approach that works for arbitrary and is taking and using the monotone convergence theorem on .