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  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CQHFPGJ45]] and let $\\mu$ be a [[Borel measure|DYR1QFNGNJ]] on $X$.\nThe *support* of $\\mu$ is the set\n$$\\operatorname{supp} \\mu = \\set{x \\in X : \\mu(N) \u003e 0 \\text{ for every open neighbourhood } N \\text{ of } x}.$$\n\n## Remarks\n\nEquivalently, the complement of $\\operatorname{supp} \\mu$ is the union of all [[open sets|02BH3MCBQB]] $U$ with $\\mu(U) = 0$.\nSo the support is always a closed set.\n\nExample: let $\\delta_0$ be the [[Dirac measure|0ACRGYNHSR]] at $0$ on $\\R$.\nEvery [[neighbourhood|0ZCY0Z80B6]] of $0$ contains $0$, so it has measure $1$, and $0 \\in \\operatorname{supp} \\delta_0$.\n\nNon-example: the point $x = 1$ is not in $\\operatorname{supp} \\delta_0$.\nThe open neighbourhood $(1/2, 3/2)$ of $1$ does not contain $0$, so its measure is $0$.\nThus $\\operatorname{supp} \\delta_0 = \\set{0}$.\n\nFor Lebesgue measure on $\\R^n$, the support is all of $\\R^n$.\nIf $X$ is second countable, or if $\\mu$ is a Radon measure, then $\\mu(X \\setminus \\operatorname{supp} \\mu) = 0$.\n\n## Explanations\n\n### L1 human\n\nThe support collects the points where the measure has mass in every small region around them.\nRemoving the open regions of zero mass leaves exactly the support.\n",
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