{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $n \\ge 1$, let $\\lambda$ be [[Lebesgue measure|2Z2QVWZ262]] on $\\R^n$, and let $A \\subseteq \\R^n$ be Lebesgue measurable.\nFor $x \\in \\R^n$ and $r \u003e 0$, let $B(x, r)$ be the [[open ball|0YE5868HZB]] of centre $x$ and radius $r$.\nThen for [[almost every|RARTMA0ZEC]] $x \\in A$,\n$$\\lim_{r \\to 0^+} \\frac{\\lambda(A \\cap B(x, r))}{\\lambda(B(x, r))} = 1,$$\nand for almost every $x \\in \\R^n \\setminus A$ the same limit equals $0$.\n\n## Remarks\n\nApply the Lebesgue differentiation theorem to the indicator function $\\mathbf{1}_{A \\cap B(0, R)}$ for each integer $R$.\nIts ball averages are exactly the ratios in the statement, and its values are $1$ on $A \\cap B(0,R)$ and $0$ off it.\n\nExample: for $A = [0, \\infty) \\subseteq \\R$, the density is $1$ at every $x \u003e 0$, $0$ at every $x \u003c 0$, and $1/2$ at $x = 0$.\nThe exceptional set $\\{0\\}$ has measure zero, as the theorem allows.\n\nThe theorem says that a measurable set cannot be spread out evenly at small scales: near almost every one of its points it fills almost all of a small ball.\nFor instance, no measurable $A \\subseteq [0,1]$ can satisfy $\\lambda(A \\cap I) = \\lambda(I)/2$ for every interval $I \\subseteq [0,1]$.\n\n## Explanations\n\n### L1 human\n\nPick a measurable set and zoom in on one of its points.\nFor almost every point of the set, the set fills almost the whole view as one zooms in.\nFor almost every point outside the set, the set almost disappears from view.\n\n### L2 human\n\nThe points where the density is $1$ are called density points.\nThe density theorem is used to show that measurable sets are, up to null sets, sets of density points, which leads to the density topology on $\\R$.\nIt can also be used to prove the Steinhaus theorem that $A - A$ contains an interval around $0$ when $\\lambda(A) \u003e 0$.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "about": [
      "2Z2QVWZ262"
    ],
    "kind": "statement",
    "msc": [
      "28A15",
      "28A75"
    ],
    "name": "Lebesgue's density theorem",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q842953",
      "wikipedia": "Lebesgue's density theorem"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Lebesgue's_density_theorem"
      }
    ],
    "tag": "WSVXYSNHNX",
    "uses": [
      "2Z2QVWZ262",
      "RARTMA0ZEC",
      "0YE5868HZB"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/WSVXYSNHNX/",
  "links": [
    {
      "field": "about",
      "target": "2Z2QVWZ262"
    },
    {
      "field": "uses",
      "target": "2Z2QVWZ262"
    },
    {
      "field": "uses",
      "target": "RARTMA0ZEC"
    },
    {
      "field": "uses",
      "target": "0YE5868HZB"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/WSVXYSNHNX.md",
  "msc": [
    "28A15",
    "28A75"
  ],
  "name": "Lebesgue's density theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/WSV/WSVXYSNHNX.md",
  "status": null,
  "tag": "WSVXYSNHNX",
  "uri": "m2://WSVXYSNHNX",
  "url": "https://math2.tamnd.com/wiki/lebesgues-density-theorem/",
  "writing": {
    "tag": "WSVXYSNHNX",
    "kind": "statement",
    "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
      }
    ]
  }
}