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  "body": "\n## Statement\n\nLet $n \\ge 1$ and let $\\lambda^*$ be Lebesgue [[outer measure|BJ314H17HC]] on [[Euclidean space|0XPR1E85W0]] $\\R^n$.\nThat is, $\\lambda^*(A)$ is the infimum of $\\sum_k \\operatorname{vol}(R_k)$ over all countable covers of $A$ by open boxes $R_k$.\nCall $E \\subseteq \\R^n$ Lebesgue measurable if for every $\\varepsilon \u003e 0$ there is an [[open set|02BH3MCBQB]] $U \\supseteq E$ with $\\lambda^*(U \\setminus E) \u003c \\varepsilon$.\nThen $E$ is Lebesgue measurable if and only if\n$$\\lambda^*(A) = \\lambda^*(A \\cap E) + \\lambda^*(A \\setminus E) \\quad \\text{for every } A \\subseteq \\R^n.$$\n\n## Remarks\n\nThe inequality $\\le$ always holds by subadditivity, so only $\\ge$ has content, and only for $A$ with $\\lambda^*(A) \u003c \\infty$.\nIf $E$ is Lebesgue measurable, approximate $E$ from outside by an open set, and use that open sets split every $A$ additively.\nConversely, apply the splitting identity to bounded parts of $E$ and to open covers chosen close to $\\lambda^*$.\nFor example, a closed box $E$ splits every $A$ additively, in line with its being measurable.\nThe criterion is useful because the splitting condition makes sense for any outer measure $\\mu^*$ on any set $X$.\nFor any outer measure, the sets $E$ with $\\mu^*(A) = \\mu^*(A \\cap E) + \\mu^*(A \\setminus E)$ for all $A \\subseteq X$ form a $\\sigma$-algebra, and $\\mu^*$ restricted to it is a complete measure.\n\n## Explanations\n\n### L1 human\n\nA set is well behaved for length, area or volume exactly when it cuts every other set cleanly.\nCutting cleanly means the outer measure of any set $A$ is the sum of the outer measures of the part of $A$ inside $E$ and the part outside $E$.\nThis test needs no topology, so it can be used in any setting where an outer measure is available.\n\n### L2 human\n\nIn a measure theory course this splitting condition is often taken as the definition of a measurable set.\nWith it, Carathéodory's construction turns an outer measure into a complete measure; this gives Lebesgue measure, Lebesgue–Stieltjes measures and Hausdorff measures.\nThe same construction is the heart of the proof of the [[Carathéodory extension theorem|Q8EXG7EKYW]].\nThe theorem above shows that for $\\R^n$ this abstract notion agrees with the geometric one of approximation by open sets.\n",
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