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  "body": "\n## Statement\n\nLet $X$ be a [[locally compact|0FE2SB8WVJ]] [[Hausdorff space|0S1FJ8VEX9]].\nLet $C_c(X)$ be the real vector space of [[continuous functions|0X3WGX95FE]] $f \\colon X \\to \\R$ with compact support.\nLet $\\Lambda \\colon C_c(X) \\to \\R$ be a [[linear form|07YQ1C78G0]] that is positive, that is, $\\Lambda f \\ge 0$ whenever $f \\ge 0$.\nThen there is a unique measure $\\mu$ on the Borel sets of $X$ with these four properties:\n\n1. $\\Lambda f = \\int_X f \\, d\\mu$ for every $f \\in C_c(X)$, where the integral is the [[Lebesgue integral|DWJF7ABPR1]].\n2. $\\mu(K) \u003c \\infty$ for every compact set $K \\subseteq X$.\n3. $\\mu(E) = \\inf \\set{\\mu(U) : U \\supseteq E, \\ U \\text{ open}}$ for every Borel set $E$.\n4. $\\mu(E) = \\sup \\set{\\mu(K) : K \\subseteq E, \\ K \\text{ compact}}$ for every open set $E$ and for every Borel set $E$ with $\\mu(E) \u003c \\infty$.\n\n## Remarks\n\nClauses 2 to 4 say that $\\mu$ is a [[Radon measure|HZNEQQBCW4]].\nThe proof sets $\\mu(U) = \\sup \\set{\\Lambda f : 0 \\le f \\le 1, \\ \\operatorname{supp} f \\subseteq U}$ for open $U$, extends it to an outer measure, and checks that the Borel sets are measurable.\nUrysohn's lemma for locally compact Hausdorff spaces supplies the functions $f$ that the construction needs.\n\nExample: on $X = \\R$, the Riemann integral $\\Lambda f = \\int_{-\\infty}^{\\infty} f(x) \\, dx$ is positive, and the measure it gives is Lebesgue measure.\nThe evaluation $\\Lambda f = f(0)$ gives the Dirac measure at $0$.\n\nLet $X$ be compact Hausdorff and give $C(X)$ the sup norm.\nA related form identifies the dual of $C(X)$ with the space of regular signed Borel measures on $X$, normed by total variation.\n\n## Explanations\n\n### L1 human\n\nA positive linear rule that assigns a number to each continuous function always comes from integration against a measure.\nSo measures and positive linear functionals on continuous functions are two descriptions of the same thing.\nThis is often used to build measures: one writes down the functional first.\n\n### L2 human\n\nThe theorem is the bridge between functional analysis and measure theory.\nIt identifies the dual of $C(X)$ for compact $X$ with a space of measures, and so gives the weak-* topology on measures that underlies weak convergence of probability measures.\nHaar measure on a locally compact group is often built by constructing an invariant positive functional and then applying this theorem.\n",
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