Assume that $h_\lambda(x)=\frac{1}{\pi} \frac{\lambda}{\lambda^2+x^2}$, for $\lambda>0$, $x \in \mathbb{R}$.
I know that if $f\in L^p$ then $\lim_{\lambda \rightarrow 0} \|f*h_\lambda -f\|_p =0$, for $1\leq p< \infty$ ( Rudin, Real and complex analysis, Thr.9.10).
How to prove that if $f\in L^1$ then $\lim_{\lambda \rightarrow 0} f*h_\lambda (x)=f(x) \textrm{ a.e. ?}$