---
tag: ZE166728M3
kind: definition
name: Potential of a measure
msc: [31C15, 31B15]
defines: JHKSF3XFY2
uses: [DYR1QFNGNJ, FQTXV6BQZG, 6TJTWQ5HCR, VDZ5FKYC2T]
sources:
  - url: https://en.wikipedia.org/wiki/Potential_theory
---

## Definition

Let $n \ge 1$ and let $k \colon \R^n \times \R^n \to (-\infty, +\infty]$ be a lower semicontinuous function, called a *kernel*.
Let $\mu$ be a positive [[Borel measure|DYR1QFNGNJ]] on $\R^n$ with finite total mass and compact [[support|FQTXV6BQZG]].
The *potential of $\mu$ with respect to $k$* is the function $U_k^\mu \colon \R^n \to (-\infty, +\infty]$,
$$U_k^\mu(x) = \int k(x, y) \, d\mu(y).$$

## Remarks

The integral is well defined: a lower semicontinuous function is bounded below on the compact set $\set{x} \times \operatorname{supp} \mu$, and $\mu$ is finite.
Example: for the Dirac measure $\mu = \delta_a$ one gets $U_k^\mu(x) = k(x, a)$.
Example: with the [[Newtonian kernel|6TJTWQ5HCR]] on $\R^3$ and $\mu$ the uniform probability measure on the unit sphere, $U_k^\mu(x) = 1$ for $\abs{x} \le 1$ and $1/\abs{x}$ for $\abs{x} \ge 1$.
Non-example: $k(x, y) = -1/\abs{x - y}$ on $\R^3$ off the diagonal is not a kernel in this sense.
It tends to $-\infty$ at the diagonal, so no value in $(-\infty, +\infty]$ on the diagonal makes it lower semicontinuous.
The potential $U_k^\mu$ is itself lower semicontinuous, by Fatou's lemma.
Standard kernels are the Newtonian kernel, the [[logarithmic kernel|VDZ5FKYC2T]] and the Riesz kernels $\abs{x - y}^{\alpha - n}$ with $0 < \alpha < n$.
Signed measures are handled by writing $\mu = \mu^+ - \mu^-$ where both potentials are finite.