Prove the series converges almost everywhere

Question: Given Lebesgue integrable $f: mathbb{R}rightarrow [0,infty)$, prove the following series converges almost everywhere on $mathbb{R}$:
$$varphi(x) = lim_{krightarrow infty} sum_{t=-k}^k f(t+x)$$

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 $f$.