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Let $1\leq p \le \infty$. If $\{f_n\}$ is a sequence in $L^p$ that converges to $f$ in $L^p$, how can I show that there exists a subsequence, say, $f_{n_k}$ that converges pointwise a.e. to $f$. (without using convergence in measure)?

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    [Related](http://math.stackexchange.com/q/138043/8271)2012-05-02

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