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  "body": "\n## Definition\n\nLet $n \\ge 2$ and let $Z \\subseteq \\R^n$.\nThe set $Z$ is *polar* if there is a [[subharmonic function|Z1MHMGFNFW]] $u$ on $\\R^n$, not identically $-\\infty$, such that\n$$Z \\subseteq \\set{ x \\in \\R^n : u(x) = -\\infty }.$$\n\n## Remarks\n\nExample: in $\\R^2$, the function $u(x) = \\log \\abs{x - a}$ is subharmonic, not identically $-\\infty$, and equals $-\\infty$ exactly at $a$, so $\\set{a}$ is polar.\nIn $\\R^n$ with $n \\ge 3$ the function $u(x) = -\\abs{x - a}^{2-n}$ plays the same role.\nA countable union of polar sets is polar, so every countable subset of $\\R^n$ is polar.\n\nNon-example: the segment $[0, 1] \\times \\set{0}$ in $\\R^2$ is not polar, because it has positive logarithmic capacity $1/4$.\n\nPolarity is a local property.\nA set $Z$ is polar if and only if each point of $Z$ has a neighbourhood $U$ with the following property.\nSome subharmonic function on $U$, not identically $-\\infty$ on any component, equals $-\\infty$ on $Z \\cap U$.\nFor Borel sets, being polar is equivalent to having capacity zero.\nA polar set has Lebesgue measure zero.\n",
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