{
  "aliases": null,
  "backlinks": [
    {
      "field": "about",
      "target": "3WJFWAGKXR"
    },
    {
      "field": "about",
      "target": "6K8GPMHWBB"
    },
    {
      "field": "about",
      "target": "QTX0GBZMT8"
    },
    {
      "field": "about",
      "target": "RJK6ZKNF6R"
    },
    {
      "field": "about",
      "target": "W6SXM59TS4"
    },
    {
      "field": "uses",
      "target": "3WJFWAGKXR"
    },
    {
      "field": "uses",
      "target": "6K8GPMHWBB"
    },
    {
      "field": "uses",
      "target": "86PVRZ08VK"
    },
    {
      "field": "uses",
      "target": "FH0Z9VHRH7"
    },
    {
      "field": "uses",
      "target": "MVYTPJ3V05"
    },
    {
      "field": "uses",
      "target": "QTX0GBZMT8"
    },
    {
      "field": "uses",
      "target": "RJK6ZKNF6R"
    },
    {
      "field": "uses",
      "target": "TS9TBY8ES1"
    },
    {
      "field": "uses",
      "target": "W6SXM59TS4"
    },
    {
      "field": "uses",
      "target": "WWVK5WF6WY"
    },
    {
      "field": "defines",
      "target": "GT9E1NT6KC"
    }
  ],
  "body": "\n## Summary\n\nA polar set in $\\R^n$, $n \\ge 2$, is a set on which some subharmonic function, not identically $-\\infty$, takes the value $-\\infty$.\nPolar sets are the negligible sets of [[potential theory|9PM8Q2E30Q]].\nA Borel set is polar exactly when it has [[capacity|4YZVFGJVSZ]] zero, Newtonian capacity for $n \\ge 3$ and [[logarithmic capacity|DBQXRTC0VN]] for $n = 2$.\nSuch sets are removable for bounded harmonic functions, and Brownian motion almost surely never hits them.\n\n## Explanations\n\n### L1 human\n\nA [[subharmonic function|Z1MHMGFNFW]] may take the value $-\\infty$ at some points, like $\\log \\abs{z}$ at $0$ in the plane.\nA polar set is a set that is small enough to be sent to $-\\infty$ by one subharmonic function at once.\nPoints and countable sets are polar.\nA line segment in the plane is not polar, even though it has area zero.\nSo polar sets are much smaller than sets of measure zero.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "31B15",
      "31A15",
      "31C15"
    ],
    "name": "Polar set",
    "same_as": {
      "wikidata": "Q7209102",
      "wikipedia": "Polar set (potential theory)"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Polar_set_(potential_theory)"
      }
    ],
    "tag": "C95QJ7YH5T"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/C95QJ7YH5T/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/C95QJ7YH5T.md",
  "msc": [
    "31B15",
    "31A15",
    "31C15"
  ],
  "name": "Polar set",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/C95/C95QJ7YH5T.md",
  "status": null,
  "tag": "C95QJ7YH5T",
  "uri": "m2://C95QJ7YH5T",
  "url": "https://tamnd.github.io/math2/wiki/polar-set/",
  "writing": {
    "tag": "C95QJ7YH5T",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}