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  "body": "\n## Definition\n\nLet $X$ be a set and let $\\mathcal{R}$ be a [[ring of sets|KWE1DBMN9B]] on $X$.\nA *premeasure* on $\\mathcal{R}$ is a map $\\mu_0 \\colon \\mathcal{R} \\to [0, \\infty]$, with values in the [[extended real line|0GSZ7B0CGR]], such that:\n\n1. $\\mu_0(\\varnothing) = 0$.\n2. For every sequence $A_1, A_2, \\ldots$ of pairwise disjoint sets of $\\mathcal{R}$ whose union $\\bigcup_n A_n$ lies in $\\mathcal{R}$,\n$$\\mu_0\\Big(\\bigcup_{n=1}^{\\infty} A_n\\Big) = \\sum_{n=1}^{\\infty} \\mu_0(A_n).$$\n\n## Remarks\n\nExample: on the ring of finite unions of half-open intervals $(a, b]$ in $\\R$, the total length is a premeasure; countable additivity is checked with a compactness argument on closed subintervals.\nNon-example: on the ring of finite and cofinite subsets of $\\N$, put $\\mu_0(A) = 0$ for finite $A$ and $\\mu_0(A) = 1$ for cofinite $A$.\nIt is finitely additive, but $\\N = \\bigcup_n \\set{n}$ gives $1 \\ne \\sum_n 0$, so clause 2 fails.\nA premeasure is monotone on $\\mathcal{R}$, and countably subadditive for sequences in $\\mathcal{R}$ whose union lies in $\\mathcal{R}$.\nThe same definition is used for a semiring of sets instead of a ring.\nA measure is a premeasure whose domain is a $\\sigma$-algebra.\n",
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