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  "body": "\n## Definition\n\nLet $E$ be a real Hausdorff [[locally convex space|B6GY2KJMWW]], and let $K \\subseteq E$ be a nonempty [[compact|0R6EWFKRCF]] [[convex set|02AS45T3H9]].\nLet $\\mu$ be a Borel [[probability measure|059TJN0NZ1]] on $K$, and let $x \\in K$.\nThe measure $\\mu$ *represents* $x$, and $x$ is the *barycenter* of $\\mu$, if\n$$\\ell(x) = \\int_K \\ell \\, d\\mu$$\nfor every [[continuous linear form|03TM135G0N]] $\\ell \\colon E \\to \\R$.\n*Choquet theory* is the study of representing measures of points of $K$ that are concentrated on the set $\\operatorname{ext} K$ of [[extreme points|3MSGFJM2NX]] of $K$.\n\n## Remarks\n\nEvery point of $K$ is represented by the [[Dirac measure|0ACRGYNHSR]] at that point; the theory asks for measures on the extreme points.\nChoquet's theorem: if $K$ is metrizable, then $\\operatorname{ext} K$ is a $G_\\delta$ set, and every $x \\in K$ is represented by a probability measure $\\mu$ with $\\mu(\\operatorname{ext} K) = 1$.\nThe Bishop–de Leeuw theorem extends this to non-metrizable $K$ in a weaker form.\nFor metrizable $K$, the measure on $\\operatorname{ext} K$ is unique for every $x$ exactly when $K$ is a Choquet simplex.\n\nExample: let $K = [0, 1] \\subseteq \\R$, with $\\operatorname{ext} K = \\set{0, 1}$.\nThe measure $\\mu = (1 - x) \\delta_0 + x \\delta_1$ represents $x$.\nThe check: for $\\ell(t) = c t$, one gets $\\int \\ell \\, d\\mu = c x = \\ell(x)$.\n\nNon-example: for $x = 1/2$, the measure $\\delta_0$ is concentrated on $\\operatorname{ext} K$ but does not represent $x$, since its barycenter is $0$.\n\nFor the square $K = [0, 1]^2$, the centre is represented by $\\frac12(\\delta_{(0,0)} + \\delta_{(1,1)})$ and by $\\frac12(\\delta_{(1,0)} + \\delta_{(0,1)})$, so uniqueness fails.\n\n## Explanations\n\n### L1 human\n\nA representing measure writes a point as an average of other points of the set.\nChoquet theory asks for averages of extreme points only, the corners of the set.\nIt is the infinite-dimensional version of barycentric coordinates.\n",
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