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  "body": "\n## Statement\n\nLet $X$ be a set and let $\\mathcal{S}$ be a [[semiring of sets|PYH82FMXQ2]] on $X$.\nLet $\\mu_0 \\colon \\mathcal{S} \\to [0, \\infty]$ be a [[premeasure|PV0XXSWFC7]], that is, $\\mu_0(\\varnothing) = 0$ and $\\mu_0\\big(\\bigcup_n A_n\\big) = \\sum_n \\mu_0(A_n)$ for every sequence of pairwise disjoint sets $A_n \\in \\mathcal{S}$ whose union lies in $\\mathcal{S}$.\nLet $\\sigma(\\mathcal{S})$ be the [[sigma-algebra|0SV8K0TDGY]] generated by $\\mathcal{S}$.\n\n1. There is a [[measure|PR11BZQQPG]] $\\mu$ on $\\sigma(\\mathcal{S})$ with $\\mu(A) = \\mu_0(A)$ for all $A \\in \\mathcal{S}$.\n2. If moreover $X = \\bigcup_{n} X_n$ for a sequence of sets $X_n \\in \\mathcal{S}$ with $\\mu_0(X_n) \u003c \\infty$, then this measure $\\mu$ is unique.\n\n## Remarks\n\nFor existence, define $\\mu^*(B) = \\inf \\set{ \\sum_n \\mu_0(A_n) : A_n \\in \\mathcal{S}, \\ B \\subseteq \\bigcup_n A_n }$.\nThis is an outer measure, and every set of $\\mathcal{S}$ satisfies the Carathéodory splitting condition.\nCountable additivity of $\\mu_0$ gives $\\mu^* = \\mu_0$ on $\\mathcal{S}$.\nSo $\\mu^*$ restricted to the Carathéodory-measurable sets is the required extension.\nUniqueness follows from Dynkin's $\\pi$-$\\lambda$ theorem, since $\\mathcal{S}$ is closed under finite intersections.\nExample: the half-open intervals $(a, b]$ form a semiring on $\\R$, and $\\mu_0((a, b]) = b - a$ is a premeasure; the theorem gives Lebesgue measure on the Borel sets, and it is unique.\nUniqueness can fail without the finiteness hypothesis in clause 2.\nOn $\\Q$, take the semiring of sets $(a, b] \\cap \\Q$ and the premeasure that is $\\infty$ on nonempty sets and $0$ on $\\varnothing$.\nIt extends both to counting measure and to $2$ times counting measure on the power set of $\\Q$.\n\n## Explanations\n\n### L1 human\n\nIt is often easy to say how big simple sets are, such as intervals or boxes.\nThe theorem says that if this size is countably additive on the simple sets, it can be extended to a measure on every set built from them by countable operations.\nIf the whole space is covered by countably many simple sets of finite size, there is only one way to do it.\n\n### L2 human\n\nThis is the main tool to construct measures: Lebesgue measure, Lebesgue–Stieltjes measures, product measures and measures on sequence spaces all come from it.\nThe only work left in each case is to check that the given set function is countably additive on the semiring.\nA version for an algebra of sets, with the extension given by an explicit outer measure, is the [[Hahn–Kolmogorov theorem|FN2WRRF6V9]].\nThe proof goes through an [[outer measure|BJ314H17HC]] and [[Carathéodory's criterion|JDGQ135YK2]].\n",
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