Choquet capacity

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.