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  "body": "\n## Statement\n\nLet $n \\ge 2$ and let $\\Omega \\subseteq \\R^n$ be a bounded [[open set|02BH3MCBQB]].\nCall a point of the [[boundary|05D5EGMFYV]] $\\partial\\Omega$ irregular if it is not a [[regular boundary point|B7948M365K]].\nThen the set of irregular points of $\\partial\\Omega$ is a [[polar set|C95QJ7YH5T]].\n\n## Remarks\n\nPolar sets have capacity zero, so they are very small: they have Lebesgue measure zero, and in $\\R^n$ with $n \\ge 3$ they have Hausdorff dimension at most $n - 2$.\nConsequence: for each continuous $f$ on $\\partial\\Omega$, the Perron–Wiener–Brelot solution $H_f$ tends to $f(\\xi)$ at every boundary point $\\xi$ outside a polar set, that is, quasi-everywhere.\nSo the generalized solution solves the Dirichlet problem up to an exceptional set of capacity zero.\nExample: for the punctured ball, the irregular set is $\\set{0}$, a single point, which is polar.\nExample: for the [[Lebesgue spine|EFFC7688KR]], the only irregular point is the tip of the spike.\n\n## Explanations\n\n### L1 human\n\nSome boundary points can be bad for the Dirichlet problem, but there are always very few of them.\nThey form a set so thin that electric charge cannot sit on it.\nSo the generalized solution takes the right boundary values almost everywhere, in the sense of capacity.\n\n### L2 human\n\nThe theorem turns the Perron–Wiener–Brelot solution into a genuine solution off a negligible set.\nIt is the basis for treating boundary values quasi-everywhere in modern potential theory and in the theory of Sobolev spaces.\nIts proof uses the fine topology or the equilibrium potential of compact parts of the boundary, together with the barrier characterization of regular points.\n",
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