# Approximation of subharmonic functions

Let $u$ be an (upper semi-continuous) locally bounded subharmonic function in a domain in $\mathbb{R}^n$. Let $\chi_\epsilon$ be a standard smoothing kernel, namely $$\chi_\epsilon(x)=\frac{c_n}{\varepsilon^n}\chi(\varepsilon^{-1} ||x||),$$ where $\chi\colon \mathbb{R}\to \mathbb{R}_{\geq 0}$ is a smooth non-negative compactly supported function, $||\cdot ||$ is the Euclidean norm, and $c_n$ is a normalizing constant such that $\int_{\mathbb{R}^n}\chi_\varepsilon(x) dx=1$.

Let $u_\varepsilon:=u\ast \chi_\varepsilon$.

Question. Is it true that $u_\varepsilon\to u$ pointwise as $\varepsilon \to 0$?

This is obviously true if $u$ is continuous.

A reference would be very helpful.

Sketch of the proof. Let us prove this at $0$. Let $m(r)=(1/2\pi)\int_{-\pi}^\pi u(re^{it})dt$. Then $m$ is continuous and $m(r)\to u(0)$ as $r\to 0$.
Whether $\chi$ is smooth or just continuous, does not matter.