---
tag: 1Z68SX12PA
kind: definition
name: Energy of a measure
msc: [31B15, 31A15]
defines: SFSRQCM3EG
uses: [DYR1QFNGNJ, FQTXV6BQZG, 6TJTWQ5HCR, VDZ5FKYC2T, JHKSF3XFY2, 4YZVFGJVSZ]
sources:
  - url: https://en.wikipedia.org/wiki/Capacity_of_a_set
---

## Definition

Let $n \ge 2$ and let $\mu$ be a finite positive [[Borel measure|DYR1QFNGNJ]] on $\R^n$ with compact [[support|FQTXV6BQZG]].
Let $k$ be the [[Newtonian kernel|6TJTWQ5HCR]] $k(x) = \abs{x}^{2-n}$ when $n \ge 3$, and the [[logarithmic kernel|VDZ5FKYC2T]] $k(x) = \log(1/\abs{x})$ when $n = 2$, with $k(0) = +\infty$.
The *energy* of $\mu$ is
$$I(\mu) = \iint k(x - y)\, d\mu(x)\, d\mu(y).$$
For $n \ge 3$ it lies in $[0, +\infty]$; for $n = 2$ it lies in $(-\infty, +\infty]$, because $k(x - y)$ is bounded below on the compact set $\operatorname{supp}\mu \times \operatorname{supp}\mu$.

## Remarks

Equivalently, $I(\mu) = \int U^\mu \, d\mu$, where $U^\mu(x) = \int k(x - y)\, d\mu(y)$ is the [[potential|JHKSF3XFY2]] of $\mu$.
Some authors include the normalising constant of the fundamental solution in $k$, which multiplies the energy by a constant.
The infimum of $I(\mu)$ over probability measures on a compact set $K$ determines the [[capacity|4YZVFGJVSZ]] of $K$.
For instance, when $n \ge 3$ and with this kernel, the Newtonian capacity of $K$ is the reciprocal of that infimum.

Example: let $\sigma$ be the uniform probability measure on the unit sphere in $\R^3$.
Its potential is $U^\sigma(x) = 1/\max(\abs{x}, 1)$, which equals $1$ on the sphere, so $I(\sigma) = 1$.
In the plane, the uniform probability measure on the unit circle has potential $\log(1/\max(\abs{z}, 1))$, which is $0$ on the circle, so its energy is $0$.

Non-example: the Dirac measure $\delta_0$ is a positive Borel measure with compact support, but $I(\delta_0) = k(0) = +\infty$, so it does not have finite energy.