Polar set

Remarks

Example: in $\R^2$, the function $u(x) = \log \abs{x - a}$ is subharmonic, not identically $-\infty$, and equals $-\infty$ exactly at $a$, so $\set{a}$ is polar. In $\R^n$ with $n \ge 3$ the function $u(x) = -\abs{x - a}^{2-n}$ plays the same role. A countable union of polar sets is polar, so every countable subset of $\R^n$ is polar.

Non-example: the segment $[0, 1] \times \set{0}$ in $\R^2$ is not polar, because it has positive logarithmic capacity $1/4$.

Polarity is a local property. A set $Z$ is polar if and only if each point of $Z$ has a neighbourhood $U$ with the following property. Some subharmonic function on $U$, not identically $-\infty$ on any component, equals $-\infty$ on $Z \cap U$. For Borel sets, being polar is equivalent to having capacity zero. A polar set has Lebesgue measure zero.