You can use the MCT for decreasing functions, since is integrable.
The exercise is to compute the limit below if it exist. I know that I'm supposed to try to use the Monotone Conv. Thm or the Dominated Conv. Thm.
First i see that the MCT is not going to work here since , where . So I guess I'm left with the DCT. But here I can't find an such that a.e. (*) nor can I find an in such that . All I know for sure is and this function is not in .
(*) actually I guess that since except on a countable set.
What to do?