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  "body": "\n## Definition\n\nLet $\\Omega \\subset \\C$ be a bounded [[domain|097KNJFJ7V]] with [[boundary|0JJ9J1XZ8G]] $\\partial\\Omega$.\nAssume that the [[Dirichlet problem|8D56GQ4AG8]] on $\\Omega$ is solvable for continuous data.\nThat is, for every continuous $\\varphi \\colon \\partial\\Omega \\to \\R$ there is a continuous $u_\\varphi \\colon \\overline{\\Omega} \\to \\R$ that is [[harmonic|0W5NJQ8MKY]] on $\\Omega$ and equals $\\varphi$ on $\\partial\\Omega$.\nBy the maximum principle, $u_\\varphi$ is unique.\nLet $z \\in \\Omega$.\nThe *harmonic measure* of $\\Omega$ at $z$ is the unique Borel probability measure $\\omega_z = \\omega(z, \\cdot, \\Omega)$ on $\\partial\\Omega$ such that\n$$u_\\varphi(z) = \\int_{\\partial\\Omega} \\varphi \\, d\\omega_z \\qquad \\text{for every continuous } \\varphi \\colon \\partial\\Omega \\to \\R.$$\nFor a [[Borel set|0BVHZXWKCD]] $E \\subseteq \\partial\\Omega$, the number $\\omega(z, E, \\Omega) = \\omega_z(E)$ is the harmonic measure of $E$ at $z$.\n\n## Remarks\n\nThe measure exists by the Riesz representation theorem.\nIndeed, by the maximum principle, $\\varphi \\mapsto u_\\varphi(z)$ is a positive linear functional on the continuous functions on $\\partial\\Omega$, and it sends $1$ to $1$.\nFor fixed $E$, the function $z \\mapsto \\omega(z, E, \\Omega)$ is harmonic on $\\Omega$ with values in $[0, 1]$.\nExample: for the unit disc $\\mathbb{D}$, the [[Poisson kernel|TCYSWCTYYM]] gives $d\\omega_z(e^{i\\theta}) = \\frac{1 - \\abs{z}^2}{2\\pi \\abs{e^{i\\theta} - z}^2} \\, d\\theta$.\nAt $z = 0$ this is $d\\theta / 2\\pi$, so the harmonic measure of an arc is its length divided by $2\\pi$.\nNon-example: the punctured disc $\\mathbb{D} \\setminus \\set{0}$ does not satisfy the hypothesis.\nThe data $1$ at $0$ and $0$ on the unit circle have no harmonic solution that is continuous up to the boundary.\nWith the more general Perron solution, the point $0$ gets harmonic measure $0$.\nIf $f \\colon \\Omega \\to \\Omega'$ is a conformal map that extends to a homeomorphism of the closures, then $\\omega(f(z), f(E), \\Omega') = \\omega(z, E, \\Omega)$.\nProbabilistically, $\\omega(z, E, \\Omega)$ is the probability that a Brownian motion started at $z$ first leaves $\\Omega$ through $E$.\n",
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