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  "body": "\n## Definition\n\nLet $X$ be a topological space and let $\\mu$ be a [[Borel measure|DYR1QFNGNJ]] on $X$.\nThe measure $\\mu$ is *locally finite* if every point $x \\in X$ has an [[open|02BH3MCBQB]] neighbourhood $U$ with $\\mu(U) \u003c \\infty$.\n\n## Remarks\n\nExample: Lebesgue measure $\\lambda$ on $\\R^n$ is locally finite.\nThe open ball of radius $1$ around $x$ has finite measure.\n\nNon-example: counting measure on $\\R$ is not locally finite.\nEvery open neighbourhood of a point is infinite, so it has measure $\\infty$.\n\nA locally finite measure is finite on every [[compact set|0R6EWFKRCF]] $K$: finitely many neighbourhoods of finite measure cover $K$.\nConversely, if $X$ is a Hausdorff [[locally compact space|0FE2SB8WVJ]] and $\\mu$ is finite on compact sets, then $\\mu$ is locally finite, since each point has a compact neighbourhood.\n",
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