Let $f \in L^1( \mathbb{R}^n)$. Does the sum $$S(x) = \sum_{k \in \mathbb{Z}^n} f(k+x) $$ converge for almost every $x$? Intuitively I'm approximating the integral (which is finite), so I think this should be true. Maybe $C||f||_{L^1}$ can be a valid upperbound for some $C>0$?
I think we can suppose wlog $n=1$ (because we can iterate) and $f\geq 0$ (because we can split $f$ in its positive and negative parts)
For $n=1$ consider $ \sum_k \int_{[0,1)} |f(x+k)| dx=\sum_k \int_{[k,k+1)} |f(x)| dx=\int_{\mathbb R} |f(x)|dx$. By Tonelli's theorem we can write this as $ \int_{[0,1)} \sum_k|f(x+k)| dx=\int_{\mathbb R} |f(x)|dx$. Hence $\sum_k|f(x+k)| <\infty$ almost everywhere on $[0,1)$. By periodcity of the sum this implies that the series converges absolutely almost everywhere on $\mathbb R$.
Almost identical argument works in $\mathbb R^{n}$. Just replace $[0,1)$ by $[0,1)\times [0,1)\times ...\times [0,1)$