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  "body": "\n## Definition\n\nLet $(X, \\Sigma)$ be a [[measurable space|N4KDT83V3S]].\nA [[measure|PR11BZQQPG]] $\\mu$ on $(X, \\Sigma)$ is *s-finite* if there are [[finite measures|63Z1Q4FS23]] $\\mu_1, \\mu_2, \\ldots$ on $(X, \\Sigma)$ such that $\\mu(A) = \\sum_{n=1}^{\\infty} \\mu_n(A)$ for every $A \\in \\Sigma$.\n\n## Remarks\n\nLet $\\lambda$ be Lebesgue measure on the Borel sets of $\\R$.\nLet $\\mu(A) = \\sum_{n=1}^{\\infty} \\lambda(A \\cap [0, 1])$, which is $0$ when $\\lambda(A \\cap [0, 1]) = 0$ and $\\infty$ otherwise.\nIt is s-finite, since each term is a finite measure.\nIt is not [[σ-finite|WEBBT18D4X]]: a set of finite $\\mu$-measure meets $[0, 1]$ in a $\\lambda$-null set, and countably many such sets cannot cover $[0, 1]$.\nEvery σ-finite measure is s-finite: write $X$ as a disjoint union of sets $B_n$ of finite measure, and let $\\mu_n(A) = \\mu(A \\cap B_n)$.\nCounting measure on the Borel sets of $\\R$ is not s-finite.\nA finite measure gives positive mass to at most countably many points.\nSo countably many finite measures give positive mass to at most countably many points in total.\nCounting measure gives mass $1$ to every point of the uncountable set $\\R$.\n",
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