What is the counterexample to show that almost sure convergence does not imply mean (L^p space) convergence?
All the examples I've seen were about the fact that convergence in mean does not imply almost sure convergence. I have not found any counterexamples for vice versa, I really want to prove it because if I find a counterexample, than I will automatically prove that measure convergence do not imply convergence in mean.
Also I do not know what Borel-Cantelli lemma is and would like to understand an example without it