Almost everywhere convergence of convolution with mollifiers

Question

I read that for $j\in L^1({\mathbb R}^n)$ with $\|j\|_1=1$ and $f\in L^1_{\rm loc}({\mathbb R}^n)$ the mollifiers $j_\epsilon(x):=\epsilon^{-n}j(x/\epsilon)$ exhibit $j_\epsilon\ast f\in L^1({\mathbb R}^n)$ and $j_\epsilon\ast f\rightarrow f$ in $L^1({\mathbb R}^n)$ as $\epsilon\rightarrow 0$. But convergence in $L^1({\mathbb R}^n)$ does not imply convergence almost everywhere, yet I couldn't find a counterexample for this case, so do we additionally have $j_\epsilon\ast f\rightarrow f$ pointwise almost everywhere or not?