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  "body": "\n## Definition\n\nLet $K$ be a [[compact set|0R6EWFKRCF]] in $\\C$.\nFor a Borel [[probability measure|059TJN0NZ1]] $\\mu$ on $K$, let\n$$I(\\mu) = \\int_K \\int_K \\log\\frac{1}{\\abs{z - w}} \\, d\\mu(z) \\, d\\mu(w)$$\nbe its logarithmic [[energy|SFSRQCM3EG]], with values in $(-\\infty, \\infty]$.\nLet $V(K) = \\inf_\\mu I(\\mu)$, the infimum over all Borel probability measures $\\mu$ on $K$.\nAssume that $V(K) \u003c \\infty$, that is, $K$ has positive [[logarithmic capacity|DBQXRTC0VN]].\nAn *equilibrium measure* of $K$ is a Borel probability measure $\\nu$ on $K$ with $I(\\nu) = V(K)$.\n\n## Remarks\n\nSuch a measure exists, because the probability measures on $K$ form a weak-star compact set and $I$ is lower semicontinuous for this topology.\nIt is unique, because the energy is strictly convex on probability measures of finite energy.\nSo one speaks of *the* equilibrium measure of $K$.\n\nExample: let $K$ be the unit circle and let $\\nu$ be normalized arc length $d\\theta / 2\\pi$.\nFor $\\abs{z} \\le 1$ the mean value $\\frac{1}{2\\pi} \\int_0^{2\\pi} \\log \\abs{z - e^{i\\theta}} \\, d\\theta$ equals $\\log \\max(\\abs{z}, 1) = 0$.\nSo the potential of $\\nu$ vanishes on $K$, and $I(\\nu) = 0 = V(K)$, which agrees with capacity $1$.\n\nNon-example: the unit point mass $\\delta_1$ is a probability measure on the unit circle, but $I(\\delta_1) = \\log(1/0) = \\infty \\ne V(K)$.\n\nIn $\\R^n$ with $n \\ge 3$ the same definition is used with the Newtonian kernel $\\abs{x - y}^{2-n}$ in place of $\\log(1/\\abs{z - w})$.\n",
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