---
tag: W9N1TR1X8P
kind: concept
name: Sets of capacity zero
msc: [31B15, 31A15, 31C15]
sources:
  - url: https://en.wikipedia.org/wiki/Capacity_of_a_set
---

## 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|DBQXRTC0VN]], and in $\R^n$ with $n \ge 3$ it is the Newtonian [[capacity|4YZVFGJVSZ]].
For Borel sets, having capacity zero is the same as being a [[polar set|C95QJ7YH5T]].
These sets are the exceptional sets of potential theory: theorems often hold everywhere except on a set of capacity zero.

## Explanations

### L1 human

Volume and area cannot tell apart a point, a curve and a surface inside space: all have volume zero.
Capacity is finer.
In space a point and a straight segment have capacity zero, but a flat disc does not.
In the plane a point and a countable set have capacity zero, but a segment does not.
Sets of capacity zero are invisible to harmonic functions that stay bounded.