---
tag: YG74XH44H6
kind: definition
name: Choquet capacity
msc: [28A12, 31C15]
defines: 25FRWWHFZV
uses: [0TVJZYB6SH, 0S1FJ8VEX9]
sources:
  - url: https://en.wikipedia.org/wiki/Capacity_of_a_set
---

## Definition

Let $X$ be a [[Hausdorff space|0S1FJ8VEX9]], and let $\mathcal{P}(X)$ be the set of all subsets of $X$.
A *Choquet capacity* on $X$ is a map $I \colon \mathcal{P}(X) \to [0, \infty]$ with the following properties.

1. If $A \subseteq B \subseteq X$, then $I(A) \le I(B)$.
2. If $A_1 \subseteq A_2 \subseteq \cdots$ are subsets of $X$ with union $A$, then $I(A_k) \to I(A)$.
3. If $K_1 \supseteq K_2 \supseteq \cdots$ are [[compact subsets|0TVJZYB6SH]] of $X$ with intersection $K$, then $I(K_k) \to I(K)$.

A set $A \subseteq X$ is *capacitable* for $I$ if $I(A) = \sup \set{I(K) : K \subseteq A, \ K \text{ compact}}$.

## Remarks

Example: let $\mu$ be a finite Borel measure on $\R^n$ and $\mu^*$ its outer measure, $\mu^*(A) = \inf \set{\mu(B) : B \supseteq A \text{ Borel}}$.
Clause 1 is clear.
Clause 2 holds because each $A_k$ has a Borel hull $B_k$ with $\mu(B_k) = \mu^*(A_k)$, and the sets $\bigcap_{j \ge k} B_j$ increase to a Borel hull of $A$.
Clause 3 is continuity from above of the finite measure $\mu$.
Non-example: on $\R$ let $I(A) = 1$ if $A$ is infinite and $I(A) = 0$ otherwise.
The finite sets $\set{1, \dots, k}$ increase to the infinite set $\N \setminus \set{0}$, but $I = 0$ along the sequence, so clause 2 fails.
Choquet's capacitability theorem: if $X$ is a Polish space, every analytic subset of $X$ is capacitable for every Choquet capacity.
Some authors allow values in $[-\infty, \infty]$ or define capacities relative to a paving other than the compact sets.