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Let $(f_n)_{n\ge1}$ be a sequence of measurable real valued functions. Prove that there exist a sequence of constants $c_n$ $>0$ such that $\sum_{i=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 $m(\{c_n |f_n| \ge 2^{-n}\}) \le 2^{-n}$. – Nate Eldredge Nov 2 '12 at 3:15
@NateEldredge: a small correction: choose $c_n$ so that $m\{|x|\le n:c_n|f_n(x)|\ge 2^{-n}\}\le 2^{-n}$. – 23rd Nov 2 '12 at 6:40
@richard why don't you post this as the answer? – Norbert Nov 2 '12 at 6:47
@Norbert: because Nate Eldredge had almost shown the answer as a comment. – 23rd Nov 2 '12 at 6:52
@richard: Thanks! Could you go ahead and post as an answer? – Nate Eldredge Nov 2 '12 at 12:33

I think this is too general a statement. If f(n)=n^m and c(n)=1/n^(m+2) , the sum will be =1/1^2+1/2^2+1/3^2+...=pi^2/6 . If c(n)=1/n^(m+s) , then the sum will be the Zeta function Z(s) --B.Sahu

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The statement is true, and the proof is basically given in the comments. Did you read them? – Lukas Geyer Nov 9 '12 at 15:38

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