Question: Given Lebesgue integrable $f: \mathbb{R}\rightarrow [0,\infty)$, prove the following series converges almost everywhere on $\mathbb{R}$: $$\varphi(x) = \lim_{k\rightarrow \infty} \sum_{t=-k}^k f(t+x)$$
Attempt: Towards a contradiction suppose there is a non-null set $A$ such that for all $x \in A$ we have $\varphi(x)=\infty$. Somehow I want to conclude that $\int_A f=\infty$ and contradict the integrability of $f$.
Let $I=(-1/2,1/2]$. Then
$$\int_{I}\Big|\sum_{k\in\mathbb{Z}}f(x+k)\,dx\Big|\leq \int_{I}\sum_{z\in\mathbb{Z}}|f(x+k)|\,dx=\sum_{k\in\mathbb{Z}}\int_{I+k}|f(x)|\,dx=\|f\|_1<\infty$$ Here the change of order of summation and integration can be justified by either monotone convegernce, or by Fubini's theorem.
Thus $g(x)=\sum_{k\in\mathbb{Z}}f(x+k)<\infty$ a.s for all $x\in I$, which can then be extended as a periodic function.
