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Let $\left(f_n\right)_{n\ge1}$ be a sequence of measurable real valued functions. Prove that there exist a sequence of constants $c_n$ $>0$ such that $\sum_{n=1}^{\infty} c_nf_n $ converges for almost every x $\in \mathbb R$.

Any hints is appreciated. Thanks

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    Try choosing $c_n$ so that $m(\{c_n |f_n| \ge 2^{-n}\}) \le 2^{-n}$.2012-11-02
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    @NateEldredge: a small correction: choose $c_n$ so that $m\{|x|\le n:c_n|f_n(x)|\ge 2^{-n}\}\le 2^{-n}$.2012-11-02
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    @richard why don't you post this as the answer?2012-11-02
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    @Norbert: because Nate Eldredge had almost shown the answer as a comment.2012-11-02
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    @richard: Thanks! Could you go ahead and post as an answer?2012-11-02
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    @NateEldredge: You are welcome. I think your hint is enough for solving this question. Please post an answer if you would like to do so.2012-11-02
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    @richard: the notation $\{f\geq c\}$ to denote $f^{-1}[c,\infty)$ is fairly standard.2012-11-02
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    Lmk if this is right, i applied borel cantelli lemma, and deduce that $c_n|f_n|<=2^{-n}$ for large n and so the series converge2012-11-04

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