Consider two sequences of real numbers $\{a_n\}$ and $\{r_n\}$ such that $\sum_{n=1}^{\infty} |a_n| < \infty.$ Prove that $$\sum_{n=1}^{\infty} \frac{a_n}{\sqrt{\strut |x-r_n|}}$$ converges absolutely for almost every $x \in \mathbb{R}$.
We are asked to give a measure-theoretic proof of this fact. We know Fubini's Theorem about interchanging the order of integration; however, I am not sure how this is supposed to help us. Could someone provide a push in the right direction or a hint as to how to solve this problem?
Hint:
$$\int_{-1}^1 \sum_{n=1}^{\infty}\frac{|a_n|}{\sqrt {|x-r_n|}}\, dx =\sum_{n=1}^{\infty} \int_{-1}^1 \frac{|a_n|}{\sqrt {|x-r_n|}}\, dx$$ $$ \le \sum_{n=1}^{\infty} \int_{-1}^1 \frac{|a_n|}{\sqrt {|x|}}\, dx = \sum_{n=1}^{\infty} 4|a_n|< \infty.$$