Let $\{F_k\}_k\subset[0,1]$ be a collection of sets such that $m(F_k)\geq \delta$ for some $\delta>0$ for all $k$. Let $\{a_k\}_k$ be a non-negative sequence such that $\sum_{k=1}^{\infty}a_k\chi_{F_k}(x)<\infty$ for almost all $x$. Show that $\sum_{k=1}^{\infty}a_k<\infty$.
I thought I could use monotone convergence theorem with $f_k=\sum_{n=1}^ka_n\chi_{F_n}$ but it didn't work.
Define $C_n:=\{x : \ \sum_k a_k \chi_{F_k}(x)<n\}$. Then almost every $x$ is contained in some $C_n$. That is, $m( \bigcup_n C_n) = 1$. Since $(C_n)$ is an increasing sequence of sets, $m( \bigcup_n C_n)=\lim_{n\to\infty} m(C_n)$. Hence there is $N$ such that $m(C_N)\ge 1-\frac\delta2$. This implies $m(C_N \cap F_k) \ge \frac\delta2$ for all $k$. And we can conclude $$ N \ge \int_{C_N} \sum_k a_k \chi_{F_k} = \sum_k a_k m(C_N \cap F_k)\ge \frac\delta 2 \sum_k a_k. $$
