Let $f: \mathbb R \to \mathbb R$ be a nonnegative measurable function, and $\{q_n\}$ some enumeration of the rational numbers. Suppose for every $0 < r < 1$ it holds that
$$\sum_{n = 0}^\infty r^n f(x + q_n)$$
converges for almost every $x \in \mathbb R$.
Question: Does this imply that for every compact set $K$ of $\mathbb R$, that $\text{esssup}_K f < \infty$?
No, this is not true. Let $f \in L^1(\mathbb R)$ and $g_r(x)=\sum_{n=0}^\infty r^n f(x+q_n) \leq \infty$. Then $\|g_r\|_1=\frac{\|f\|_1}{1-r}$ and hence $g_r(x)<\infty$ a.e. If $E_r$ is the null set where $g_r$ may not converge then the series converge for all $x$ outside $\cup_n E_{1-\frac 1n}$, for every $r<1$, since $g_r$ is increasing in $r$. Now it is sufficient to choose $f \in L^1$ but not locally bounded.
