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  "body": "\n## Definition\n\nLet $X$ be a set and let $\\mathcal{P}(X)$ be its [[power set|00E0YZCERH]].\nAn *outer measure* on $X$ is a map $\\mu^* \\colon \\mathcal{P}(X) \\to [0, \\infty]$, with values in the [[extended real line|0GSZ7B0CGR]], such that:\n\n1. $\\mu^*(\\varnothing) = 0$.\n2. If $A \\subseteq B \\subseteq X$, then $\\mu^*(A) \\le \\mu^*(B)$.\n3. For every sequence $A_1, A_2, \\ldots$ of subsets of $X$,\n$$\\mu^*\\Big(\\bigcup_{n=1}^{\\infty} A_n\\Big) \\le \\sum_{n=1}^{\\infty} \\mu^*(A_n).$$\n\n## Remarks\n\nExample: Lebesgue outer measure on $\\R$ is\n$$\\lambda^*(A) = \\inf \\Big\\{ \\sum_{n} (b_n - a_n) : A \\subseteq \\bigcup_{n} (a_n, b_n) \\Big\\},$$\nthe infimum over all countable covers of $A$ by open intervals.\nClause 1 holds since $\\varnothing$ is covered by intervals of arbitrarily small total length.\nClause 2 holds since a cover of $B$ also covers $A$.\nClause 3 follows by covering each $A_n$ within $\\varepsilon / 2^n$ of $\\lambda^*(A_n)$.\nNon-example: on $X = \\set{1, 2}$, the map with $\\mu(\\varnothing) = 0$, $\\mu(\\set{1}) = \\mu(\\set{2}) = 1$ and $\\mu(X) = 3$ is monotone, but it fails clause 3 because $3 \u003e 1 + 1$.\nSome authors, such as Federer, call outer measures simply measures.\nThe outer measures defined here are the ones to which [[Carathéodory's criterion|JDGQ135YK2]] applies.\n",
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