# Continuity of subharmonic functions

There is a result saying that the set where a subharmonic function defined on an open set of $$\mathbb{R}^{m}$$ ($$m\geq2$$) is discontinuous is a polar set. Could someone give me a reference for this result?

Let $$\Omega\subset\mathbb{R}^{2}$$ be a bounded open set, and let $$(x_{n})_{n\geq1}$$ be a sequence of all points of $$\Omega$$ with rational coordinates. Consider the discrete measure of finite mass, $$\mu=\sum_{n\geq1}\frac{1}{n^{2}}\delta_{x_{n}}.$$ Its logarithmic potential $$U^{\mu}(z)=\int\log\frac{1}{|z-t|}d\mu(t)$$ is superharmonic. It takes the value $$+\infty$$ on a polar set which contains the set of points $$x_{n}$$, $$n\geq1$$, dense in $$\Omega$$. Hence, $$U^{\mu}$$ is discontinuous at each point of $$\Omega$$ where it is finite, that is quasi-everywhere in $$\Omega$$.
However, a Lusin-type property holds : Consider a potential (or a subharmonic function) in $$\mathbb{R}^m$$. For any $$\epsilon$$ there exists an open set $$G_\epsilon$$ with capacity less than $$\epsilon$$ such that the restriction of the potential to the complement of $$G_\epsilon$$ is continuous. This is Theorem 3.6 on p.185 of Landkof's book. The proof is given for the case $$m\geq3$$ only.