Let $\phi \in L^1(\mathbb R^n)$ and $\int_{\mathbb R^n} \phi(x)dx = 1$. Let $\psi(x) = \mathrm{ess}\sup_{\vert y \vert \geq \vert x \vert} \vert \phi(y) \vert$ and assume further that $\psi \in L^1(\mathbb R^n)$. I can prove that for $f \in L^p(\mathbb R^n)$ ($1 \leq p < \infty$), then $$\lim_{t \rightarrow 0} (\phi_t * f)(x_0) = f(x_0)\ \ \ \mathrm{a.e.}$$ This can be proven trough the maximal function using the fact that $\psi$ is positive, radial, decreasing, and integrable. (This is just Corollary 2.9 of Duoandikoetxea's Fourier Analysis).
What can we say about the convergence if $x_0$ is a Lebesgue point of $f$?