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  "body": "\n## Definition\n\nLet $X$ be a [[Hausdorff space|0S1FJ8VEX9]], and let $\\mathcal{P}(X)$ be the set of all subsets of $X$.\nA *Choquet capacity* on $X$ is a map $I \\colon \\mathcal{P}(X) \\to [0, \\infty]$ with the following properties.\n\n1. If $A \\subseteq B \\subseteq X$, then $I(A) \\le I(B)$.\n2. If $A_1 \\subseteq A_2 \\subseteq \\cdots$ are subsets of $X$ with union $A$, then $I(A_k) \\to I(A)$.\n3. If $K_1 \\supseteq K_2 \\supseteq \\cdots$ are [[compact subsets|0TVJZYB6SH]] of $X$ with intersection $K$, then $I(K_k) \\to I(K)$.\n\nA set $A \\subseteq X$ is *capacitable* for $I$ if $I(A) = \\sup \\set{I(K) : K \\subseteq A, \\ K \\text{ compact}}$.\n\n## Remarks\n\nExample: let $\\mu$ be a finite Borel measure on $\\R^n$ and $\\mu^*$ its outer measure, $\\mu^*(A) = \\inf \\set{\\mu(B) : B \\supseteq A \\text{ Borel}}$.\nClause 1 is clear.\nClause 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 $A$.\nClause 3 is continuity from above of the finite measure $\\mu$.\nNon-example: on $\\R$ let $I(A) = 1$ if $A$ is infinite and $I(A) = 0$ otherwise.\nThe 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.\nChoquet's capacitability theorem: if $X$ is a Polish space, every analytic subset of $X$ is capacitable for every Choquet capacity.\nSome authors allow values in $[-\\infty, \\infty]$ or define capacities relative to a paving other than the compact sets.\n",
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