# Is this a superharmonic function?

Hi everyone: Let $\Omega$ be a bounded open set of $\mathbb{R}^{N}$, $N\geq2$, and $F\subset \Omega$ with empty interior. Suppose there exists a superharmonic function $u$ on $\Omega\setminus F$ such that $$\lim_{x\rightarrow y}u(x)=+\infty$$ for all $y\in F$. Now, we define $w(x)$ to be equal to $u(x)$ for $x\in \Omega\setminus F$, and $+\infty$ for $x\in F$. Is $w$ a superharmonic function on $\Omega$?

First you have to assume that $F$ has measure zero, since a superharmonic function is locally integrable.

Under this assumption, setting $u=\infty$ on $F$ results in a superharmonic function. To see this, note that by the limit assumption $$\lim_{x\to y}u(x)=\infty$$ for all $y\in F$, the resulting function is still lower semi-continuous.

Also, the mean value property $$u(x)\geq \frac{1}{|B(x,r)|} \int_{B(x,r)} u$$ remains valid if $F$ has measure zero.

These imply that $u$ is superharmonic.

• Thanks for your answer. All I know is that the harmonic measure of $F$ with respect to $\Omega\setminus F$ is zero. But Inerrability is not a part of the definition of a superharmonic function. – M. Rahmat Nov 26 '16 at 20:55
• You are right, it is not part of the definition but it follows from the mean value property. The proof is essentially a connectedness argument that uses the fact that superharmonic functions attain a minimum on compact sets. – Dimitrios Nt Nov 26 '16 at 21:08
• I agree with what you have proved: if $F$ is a null set, then for sure $u$ is superharmonic. What I want to make sure is this: is it possible that $F$ have only empty interior without $u$ being superharmonic? – M. Rahmat Nov 26 '16 at 21:36
• @Rahmat: The answer to your question in comment is no. If $F$ has positive measure than it has positive capacity. And a superharmonic function can be infinite only on a set of zero capacity. – Alexandre Eremenko Nov 26 '16 at 21:39
• I was just thinking that the continuity of the function near $F$ makes it impossible that the measure of $F$ be positive. – M. Rahmat Nov 26 '16 at 23:08