{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "8G4ZKXPV29"
    }
  ],
  "body": "\n## Summary\n\nA doubling measure on a metric space is a Borel measure with the following property, for a fixed constant $C$.\nDoubling the radius of any ball multiplies its measure by at most $C$.\nLebesgue measure on $\\R^n$ is the model example, with constant $2^n$.\nMany tools of real analysis on $\\R^n$ extend to metric spaces with a doubling measure.\nExamples are the Vitali covering lemma, maximal function bounds and the Lebesgue differentiation theorem.\nA metric space that carries a doubling measure is a doubling metric space.\n\n## Explanations\n\n### L1 human\n\nTake any ball and look at the ball with the same centre and twice the radius.\nFor a doubling measure, the big ball never has more than $C$ times the mass of the small ball, with the same $C$ for all balls.\nSo mass is spread out without sudden thin spots next to heavy ones.\nOn the line with ordinary length, doubling an interval doubles its length, so $C = 2$ works.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "28A12",
      "30L99"
    ],
    "name": "Doubling measure",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Doubling_space"
      }
    ],
    "specializes": [
      "DYR1QFNGNJ"
    ],
    "tag": "W1JXH6NCKX"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/W1JXH6NCKX/",
  "links": [
    {
      "field": "specializes",
      "target": "DYR1QFNGNJ"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/W1JXH6NCKX.md",
  "msc": [
    "28A12",
    "30L99"
  ],
  "name": "Doubling measure",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/W1J/W1JXH6NCKX.md",
  "status": null,
  "tag": "W1JXH6NCKX",
  "uri": "m2://W1JXH6NCKX",
  "url": "https://math2.tamnd.com/wiki/doubling-measure/",
  "writing": {
    "tag": "W1JXH6NCKX",
    "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
      }
    ]
  }
}