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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.

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Yes, this is true.

References: Any book on potential theory or subharmonic functions, for example, N. Landkof, Foundations of modern potential theory, W. Hayman and P. Kennedy, Subharmonic functions I, L. Hormander, Notions of convexity.

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.

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