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  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CQHFPGJ45]] and let $\\mathcal{B}(X)$ be its [[Borel σ-algebra|0WQXY32Q8D]], the smallest σ-algebra on $X$ that contains every open set.\nA *Borel measure* on $X$ is a [[measure|PR11BZQQPG]] $\\mu \\colon \\mathcal{B}(X) \\to [0, \\infty]$.\n\n## Remarks\n\nExample: the Dirac measure $\\delta_p$ at a point $p \\in X$, with $\\delta_p(A) = 1$ if $p \\in A$ and $0$ otherwise, is a Borel measure.\nIt is defined on all subsets of $X$, hence on $\\mathcal{B}(X)$, and it is countably additive because $p$ lies in at most one set of a disjoint family.\n\nNon-example: on $\\R$, let $\\Sigma = \\set{A \\subseteq \\R : A \\text{ or } \\R \\setminus A \\text{ is countable}}$ and let $\\mu$ be $0$ on countable sets and $1$ on the others.\nThen $\\mu$ is a measure on $\\Sigma$, but $\\Sigma$ does not contain the interval $(0, 1)$, so $\\mu$ is not defined on all Borel sets and is not a Borel measure.\n\nConventions vary.\nSome authors call a measure on a σ-algebra containing $\\mathcal{B}(X)$, such as Lebesgue measure on the Lebesgue sets, a Borel measure.\nSome authors require in addition that $\\mu(K) \u003c \\infty$ for every compact set $K$; then counting measure on $\\R$ is not a Borel measure.\n",
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