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  "body": "\n## Definition\n\nLet $n \\ge 2$ and let $E \\subseteq \\R^n$.\nFor $n = 2$, identify $\\R^2$ with $\\C$ and let $\\operatorname{cap}$ be the [[logarithmic capacity|DBQXRTC0VN]].\nFor $n \\ge 3$, let $\\operatorname{cap}$ be the Newtonian [[capacity|4YZVFGJVSZ]].\nThe set $E$ has *capacity zero* if $\\operatorname{cap}(K) = 0$ for every [[compact set|0R6EWFKRCF]] $K \\subseteq E$.\n\n## Remarks\n\nThis is the condition that the inner capacity of $E$ vanishes.\nFor Borel sets it is equivalent to the vanishing of the outer capacity, by Choquet's capacitability theorem.\n\nExample: in $\\C$, a finite set $K$ has capacity zero, since every probability measure on $K$ has an atom and so has infinite logarithmic energy.\nEvery compact subset of a countable set is countable, and a countable compact set carries no atomless probability measure, so countable subsets of $\\C$ also have capacity zero.\n\nNon-example: the segment $[0, 1]$ in $\\C$ does not have capacity zero, since it is compact with logarithmic capacity $1/4$.\n\nA Borel set has capacity zero if and only if it is a [[polar set|C95QJ7YH5T]].\nIn $\\R^n$, $n \\ge 3$, a compact set with finite $(n-2)$-dimensional Hausdorff measure has capacity zero.\n",
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