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Let $(f_i)_{i\geq 1}$ be a sequence of real-valued measurable functions on $\mathbb{R}$. I want to show that there exists a sequence of real numbers $(c_j)_{j\geq 1}$, $c_j>0$ for all $j$, such that $\sum_{n,m} \lambda(E_{n,m}^{(c_j)_{j\geq 1}})$ converges, where $E_{n,m} = \{x : |\sum_{i=1}^m c_if_i (x)| > 1/n\}$ and $\lambda$ is the Lebesgue measure.

Is it possible? In general, what are obvious bounds (or bounds that are often used) on the Lebesgue measure of sets of real numbers?

Thank you!

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    Otherwise, the sum $\sum\limits_{k\geqslant n}\lambda(E_{k,m})$ diverges, hence $\sum\limits_{k}\lambda(E_{k,m})$ diverges, hence $\sum\limits_{k,\ell}\lambda(E_{k,\ell})$ diverges.2012-10-19

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