Sets of capacity zero

Summary

A set of capacity zero is a set so thin that no unit charge of finite energy can be placed on it. In the plane the relevant capacity is the logarithmic capacity, and in $\R^n$ with $n \ge 3$ it is the Newtonian capacity. For Borel sets, having capacity zero is the same as being a polar set. These sets are the exceptional sets of potential theory: theorems often hold everywhere except on a set of capacity zero.