I state the problem here:
Let $q_1, q_2, \cdots$ be an enumeration of rational numbers, and define $f: \mathbb{R} \to \mathbb{R}$ by $f(x) = \sum_{k = 1} ^\infty \frac{e^{-|x - q_k|}}{2^k \sqrt{|x - q_k|}}$. Show the function is finite almost everywhere.
I have been stuck on this problem for a while. My approach is simple: We observe that $f$ is nonnegative. Therefore, if we can somehow show $\int f d\mu < +\infty$, then the set where $f = +\infty$ is a $\mu$-null set and we are done. The problem occurs when I try to integrate $f$: \begin{align*} \int f = \int \sum_{k = 1} ^\infty \frac{e^{-|x - q_k|}}{2^k \sqrt{|x - q_k|}} = \sum_{k = 1} ^\infty \frac{1}{2^k}\int \frac{e^{-|x - q_k|}}{\sqrt{|x - q_k|}}. \end{align*} I wish to show that there is a uniform bound for $\int \frac{e^{-|x - q_k|}}{\sqrt{|x - q_k|}} \leq C$. Then as we have a geometric series, we are done. I plan to do this through partitioning the integration domain. However, it is not clear how I could separate rational point blow-ups using simple intervals. Any help is appreciated.