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  "body": "\n## Summary\n\nAn outer measure assigns a size in $[0, \\infty]$ to every subset of a set, not only to the sets of a $\\sigma$-algebra.\nIt is monotone and countably subadditive, but in general not additive.\nThrough [[Carathéodory's criterion|JDGQ135YK2]] every outer measure produces a $\\sigma$-algebra on which it is a complete [[measure|PR11BZQQPG]].\nThis is the standard way to build Lebesgue measure and Hausdorff measures.\n\n## Explanations\n\n### L1 human\n\nTo measure an odd-shaped set, cover it by simple pieces whose size is known, and take the smallest total size of such a cover.\nThis over-estimate is the outer measure.\nIt can be applied to any set at all, but for very wild sets it is not additive.\nThe union of two disjoint sets can have an outer measure smaller than the sum of their outer measures.\nFor example, Lebesgue outer measure covers a subset of $\\R$ by countably many intervals and takes the infimum of the total length.\nEvery countable set gets outer measure $0$, because the $n$-th point can be covered by an interval of length $\\varepsilon / 2^n$.\n",
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