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  "body": "\n## Definition\n\nLet $(X, \\Sigma, \\mu)$ be a [[measure space|P15VQS5DJ9]].\nThe [[measure|PR11BZQQPG]] $\\mu$ is *σ-finite* if there are sets $A_1, A_2, \\ldots$ in $\\Sigma$ such that $X = \\bigcup_{n=1}^{\\infty} A_n$ and $\\mu(A_n) \u003c \\infty$ for every $n$.\n\n## Remarks\n\nLebesgue measure on $\\R$ is σ-finite: take $A_n = [-n, n]$, of measure $2n$.\nCounting measure on the power set of $\\R$ is not σ-finite.\nA set of finite counting measure is finite, a [[countable union|01TQB88H4C]] of finite sets is countable, and $\\R$ is uncountable.\nEvery [[finite measure|63Z1Q4FS23]] is σ-finite, with $A_n = X$.\nThe sets $A_n$ can be chosen increasing, or pairwise disjoint, by the usual rearrangements.\n",
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