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  "body": "\n## Definition\n\nLet $X$ be a set and let $\\mathcal{S}$ be a [[semiring of sets|PYH82FMXQ2]] on $X$.\nMore generally, $\\mathcal{S}$ may be any family of subsets of $X$ that contains $\\varnothing$.\nA *content* on $\\mathcal{S}$ is a map $\\mu \\colon \\mathcal{S} \\to [0, \\infty]$, with values in the [[extended real line|0GSZ7B0CGR]], such that:\n\n1. $\\mu(\\varnothing) = 0$.\n2. If $A_1, \\ldots, A_n \\in \\mathcal{S}$ are pairwise disjoint and $A_1 \\cup \\cdots \\cup A_n \\in \\mathcal{S}$, then\n$$\\mu(A_1 \\cup \\cdots \\cup A_n) = \\mu(A_1) + \\cdots + \\mu(A_n).$$\n\n## Remarks\n\nExample: on the semiring of bounded intervals $[a, b)$ in $\\R$, length $\\mu([a, b)) = b - a$ is a content.\nIndeed, if $[a, b)$ is a disjoint union of finitely many such intervals, their lengths add up to $b - a$.\nNon-example: on the same semiring, $\\mu([a, b)) = (b - a)^2$ fails clause 2, since $[0, 2) = [0, 1) \\cup [1, 2)$ but $4 \\ne 1 + 1$.\nOn a ring of sets a content is monotone and finitely subadditive.\nA content on a ring is called a [[finitely additive measure|KFWQ24D9M6]] by many authors.\nA content that is countably additive is a [[premeasure|PV0XXSWFC7]].\nThe word content also names a set function used to build Haar measure.\nIt is defined on the compact subsets of a space; it is monotone and finitely subadditive, and additive on disjoint compact sets.\n",
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