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  "body": "\n## Description\n\nLet $n \\ge 1$.\nA box in $\\R^n$ is a product $R = [a_1, b_1] \\times \\cdots \\times [a_n, b_n]$ with $a_i \\le b_i$, and its volume is $\\operatorname{vol}(R) = (b_1 - a_1) \\cdots (b_n - a_n)$.\nLebesgue [[outer measure|BJ314H17HC]] on [[Euclidean space|0XPR1E85W0]] $\\R^n$ is\n$$\\lambda^*(A) = \\inf \\Big\\{ \\sum_{k=1}^{\\infty} \\operatorname{vol}(R_k) : R_1, R_2, \\ldots \\text{ boxes}, \\ A \\subseteq \\bigcup_{k} R_k \\Big\\} \\quad \\text{for } A \\subseteq \\R^n.$$\nA set $E \\subseteq \\R^n$ is Lebesgue measurable if $\\lambda^*(A) = \\lambda^*(A \\cap E) + \\lambda^*(A \\setminus E)$ for all $A \\subseteq \\R^n$.\nLebesgue measure $\\lambda$ (or $\\lambda^n$) is $\\lambda^*$ restricted to the Lebesgue measurable sets.\n\n## Remarks\n\nBy [[Carathéodory's criterion|JDGQ135YK2]] the Lebesgue measurable sets form a $\\sigma$-algebra and $\\lambda$ is a complete [[measure|PR11BZQQPG]] on it.\nEvery [[Borel set|0WQXY32Q8D]] is Lebesgue measurable, and $\\lambda(R) = \\operatorname{vol}(R)$ for every box $R$; so $\\lambda([a, b]) = b - a$ on $\\R$.\nLebesgue measure is invariant under translations and rotations, and $\\lambda(tA) = \\abs{t}^n \\lambda(A)$ for $t \\in \\R$.\nUp to a constant factor, it is the only translation-invariant measure on the Borel sets of $\\R^n$ that is finite on bounded sets.\nEvery countable set has measure $0$, and so does the Cantor set, which is uncountable.\nLebesgue measurable sets are regular: for every measurable $E$ and $\\varepsilon \u003e 0$ there are a closed set $F$ and an open set $U$ with $F \\subseteq E \\subseteq U$ and $\\lambda(U \\setminus F) \u003c \\varepsilon$.\nNot every subset of $\\R^n$ is Lebesgue measurable; a Vitali set is an example.\n\n## Explanations\n\n### L1 human\n\nLebesgue measure is length on the line, area in the plane and volume in space, extended to as many sets as possible.\nTo measure a set, cover it with countably many boxes and take the smallest possible total volume.\nThe interval $[0, 3]$ gets length $3$, a disk of radius $r$ gets area $\\pi r^2$, and the set of rational numbers gets length $0$.\n",
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