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!