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 $\text{esssup}\_K f < \infty$?



Does this condition on $f$ imply essential boundedness on compacts?
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.


