Can anybody help me, Im trying to think of an example to show that convergence in mean does not imply convergence P a.s and also to show in the other direction, i.e convergence in P a.s does not imply convergence in mean??
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Consider $\Omega=[0,1]$ with Lebesgue measure.
However, convergence in means implies that we can extract a subsequence which converges almost everywhere.