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  "body": "\n## Definition\n\nLet $(X, d)$ be a [[metric space|0VKD1TP7F3]].\nFor nonempty $A, B \\subseteq X$ let $d(A, B) = \\inf \\set{ d(a, b) : a \\in A, \\ b \\in B }$, an [[infimum|0QF1MZ9FRG]].\nA *metric outer measure* on $X$ is an [[outer measure|BJ314H17HC]] $\\mu^*$ on $X$ such that\n$$\\mu^*(A \\cup B) = \\mu^*(A) + \\mu^*(B) \\quad \\text{whenever } A, B \\subseteq X \\text{ are nonempty with } d(A, B) \u003e 0.$$\n\n## Remarks\n\nExample: Lebesgue outer measure $\\lambda^*$ on $\\R^n$ is a metric outer measure.\nLet $d(A, B) = \\delta \u003e 0$.\nA cover of $A \\cup B$ by boxes can be refined into boxes of diameter less than $\\delta$, and then no box meets both $A$ and $B$.\nSo the cover splits into a cover of $A$ and a cover of $B$, which gives $\\ge$; subadditivity gives $\\le$.\nNon-example: on $\\R$, put $\\mu^*(\\varnothing) = 0$ and $\\mu^*(A) = 1$ for every nonempty $A$.\nThis is an outer measure, but $\\mu^*(\\set{0, 1}) = 1 \\ne \\mu^*(\\set{0}) + \\mu^*(\\set{1})$ although $d(\\set{0}, \\set{1}) = 1 \u003e 0$.\nCarathéodory's theorem on metric outer measures: an outer measure on a metric space is a metric outer measure if and only if every Borel set is Carathéodory measurable for it.\n",
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