Convergence of Series of a.e. finite measurable functions Let $(f_n)_{n\geq1}$ be a sequence of measurable almost everywhere finite real-valued functions on $(X,M,\mu)$, where $µ$ is a $σ$-finite measure. Prove that there exist constants $c_n > 0$ such that the series $\sum c_nf_n(x)$ converges for $\mu$-a.e. $x\in X$
 A: Hint: Because $\mu$ is $\sigma$-finite, there is a strictly positive measurable function $g$ such that $\int g\,d\mu\le 1$.  Now choose constants $c_n>0$ so small that $\int \left[1-\exp(-c_n|f_n|)\right]g\,d\mu\le 2^{-n}$ for $n=1,2,\ldots$. The series $\sum_n c_n f_n(x)$ converges absolutely for $\mu$-a.e $x$.
Fix $n$. To see that $c_n$ exists, notice that $\lim_{c\to 0+}\int \left[1-\exp(-c|f_n|)\right]g\,d\mu=0$ by Monotone Convergence (the integrand $[1-\exp(-c|f_n|)g$ decreases to $0$ as $c$ decreases to $0$, and $\int \left[1-\exp(-|f_n|)\right]g\,d\mu\le\int g\,d\mu<\infty$).
Because $\int \left[1-\exp(-c_n|f_n|)\right]g\,d\mu\le 2^{-n}$ for each $n$ we have 
$$
\int\sum_n \left[1-\exp(-c_n|f_n|)\right]g\,d\mu=\sum_n\int \left[1-\exp(-c_n|f_n|)\right]g\,d\mu<\infty,
$$
which means that 
$$
\sum_n \left[1-\exp(-c_n|f_n|)\right]<\infty
$$
$\mu$-a.e. (because $g>0$). Let $G$ be the subset of $X$ where this series converges. Notce that if $x\in G$ then  $\lim_nc_n|f_n(x)|=0$; therefore  there exists $N(x)$ with $\mu(\{x\in G: N(x)=\infty\})=0$ such that $c_n|f_n(x)|\le 1$ for all $n\ge N(x)$. For $x\in G\cap\{N<\infty\}$ we have
$$
\eqalign{
\infty&>\sum_n \left[1-\exp(-c_n|f_n(x)|)\right]\ge\sum_{n=N(x)}^\infty \left[1-\exp(-c_n|f_n(x)|)\right]\cr
&\ge \sum_{n=N(x)}^\infty (1-e^{-1})c_n|f_n(x)|,\cr
}
$$
because $1-e^{-x}\ge (1-e^{-1})x$ for $0\le x\le 1$. Thus $\sum_{n=N(x)}^\infty c_n|f_n(x)|<\infty$ for all $x\in G\cap\{N<\infty\}$, which guarantees that $\sum_nc_n|f_n|<\infty$, $\mu$-a.e.
