{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $n \\ge 1$, let $\\lambda$ be [[Lebesgue measure|2Z2QVWZ262]] on $\\R^n$, and let $f \\colon \\R^n \\to \\R$ be locally [[integrable|X7GTF3D98F]].\nFor $x \\in \\R^n$ and $r \u003e 0$, let $B(x, r)$ be the [[open ball|0YE5868HZB]] of centre $x$ and radius $r$.\nA point $x \\in \\R^n$ is a *Lebesgue point* of $f$ if\n$$\\lim_{r \\to 0^+} \\frac{1}{\\lambda(B(x, r))} \\int_{B(x, r)} \\abs{f(y) - f(x)} \\, d\\lambda(y) = 0.$$\n\n## Remarks\n\nExample: if $f$ is continuous at $x$, then $x$ is a Lebesgue point.\nGiven $\\varepsilon \u003e 0$, choose $r_0$ with $\\abs{f(y) - f(x)} \u003c \\varepsilon$ on $B(x, r_0)$; then every average with $r \u003c r_0$ is at most $\\varepsilon$.\n\nNon-example: let $n = 1$ and $f = \\mathbf{1}_{[0, \\infty)}$.\nAt $x = 0$, the integrand equals $1$ on $(-r, 0)$ and $0$ on $[0, r)$, so the average is $1/2$ for every $r$, and $0$ is not a Lebesgue point.\n\nThe set of Lebesgue points depends on the chosen function, not only on its class modulo null sets.\nChanging $f$ at one point $x$ can destroy the Lebesgue point property at $x$ but at no other point.\n",
  "commit": "efe85cee8dbc57d85bedbacfb97b99d1b7bee843",
  "front": {
    "defines": "19WRHM5NAT",
    "kind": "definition",
    "msc": [
      "28A15"
    ],
    "name": "Lebesgue point",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Lebesgue_point"
      }
    ],
    "tag": "AWFXHJXGKE",
    "uses": [
      "2Z2QVWZ262",
      "X7GTF3D98F",
      "0YE5868HZB"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/AWFXHJXGKE/",
  "links": [
    {
      "field": "defines",
      "target": "19WRHM5NAT"
    },
    {
      "field": "uses",
      "target": "2Z2QVWZ262"
    },
    {
      "field": "uses",
      "target": "X7GTF3D98F"
    },
    {
      "field": "uses",
      "target": "0YE5868HZB"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/AWFXHJXGKE.md",
  "msc": [
    "28A15"
  ],
  "name": "Lebesgue point",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/AWF/AWFXHJXGKE.md",
  "status": null,
  "tag": "AWFXHJXGKE",
  "uri": "m2://AWFXHJXGKE",
  "url": "https://math2.tamnd.com/wiki/lebesgue-point-definition/",
  "writing": {
    "tag": "AWFXHJXGKE",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}