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  "body": "\n## Definition\n\nLet $X$ be a [[Hausdorff space|0S1FJ8VEX9]] and let $\\mu$ be a [[Borel measure|DYR1QFNGNJ]] on $X$.\nThe measure $\\mu$ is a *Radon measure* if both of the following hold.\n\n1. $\\mu$ is [[locally finite|YQ5C1SRE0W]]: every point has an open neighbourhood of finite measure.\n2. $\\mu$ is [[inner regular|28E5Q5BM69]]: $\\mu(A) = \\sup \\set{\\mu(K) : K \\subseteq A, \\ K \\text{ compact}}$ for every Borel set $A$.\n\n## Remarks\n\nExample: the Dirac measure $\\delta_p$ at a point $p$ is a Radon measure.\nEvery point has a neighbourhood of measure at most $1$; and if $p \\in A$ then $K = \\{p\\}$ is a [[compact|0R6EWFKRCF]] subset of $A$ with $\\delta_p(K) = \\delta_p(A)$, while if $p \\notin A$ both sides are $0$.\nLebesgue measure on $\\R^n$ is also a Radon measure.\n\nNon-example: counting measure on $\\R$ satisfies clause 2 but not clause 1.\n\nAnother common definition, used on locally compact Hausdorff spaces, asks that $\\mu$ be finite on compact sets, outer regular on all Borel sets and inner regular on open sets.\nOn a locally compact Hausdorff space in which every open set is σ-compact, such as $\\R^n$, every Borel measure that is finite on compact sets is a Radon measure.\n",
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