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  "body": "\n## Definition\n\nLet $(X, \\Sigma, \\mu)$ be a [[measure space|P15VQS5DJ9]].\nThe [[measure|PR11BZQQPG]] $\\mu$ is *finite* if $\\mu(X) \u003c \\infty$.\n\n## Remarks\n\nLebesgue measure on the Borel sets of $[0, 1]$ is finite, since $\\mu([0, 1]) = 1$.\nLebesgue measure on the Borel sets of $\\R$ is not finite, since $\\mu(\\R) = \\infty$.\nIf $\\mu$ is finite, then $\\mu(A) \\le \\mu(X) \u003c \\infty$ for every $A \\in \\Sigma$, by monotonicity.\nEvery [[probability measure|059TJN0NZ1]] is finite.\nIf $0 \u003c \\mu(X) \u003c \\infty$, then $\\mu / \\mu(X)$ is a probability measure.\n",
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