Polar set

Summary

A polar set in $\R^n$, $n \ge 2$, is a set on which some subharmonic function, not identically $-\infty$, takes the value $-\infty$. Polar sets are the negligible sets of potential theory. A Borel set is polar exactly when it has capacity zero, Newtonian capacity for $n \ge 3$ and logarithmic capacity for $n = 2$. Such sets are removable for bounded harmonic functions, and Brownian motion almost surely never hits them.