{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "EGQ4GHH127"
    }
  ],
  "body": "\n## Summary\n\nA [[measure|PR11BZQQPG]] is quasi-invariant under a [[group action|0WCY42EV9V]] if moving by any group element keeps the same [[null sets|VQQM4Y550T]].\nThe measure itself may change, but only by a density.\nThis weaker form of invariance is what one finds on homogeneous spaces without invariant measures and on infinite-dimensional spaces.\n\n## Explanations\n\n### L1 human\n\nAn invariant measure gives the same size to a set and to each of its moved copies.\nA quasi-invariant measure only promises that a set of size zero stays of size zero when moved.\nFor example, the bell-curve measure on the real line is not invariant under shifts, since it puts more mass near $0$.\nBut a shifted set has zero bell-curve measure exactly when it has zero length, so the measure is quasi-invariant under shifts.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "28C10",
      "28D05",
      "22D40"
    ],
    "name": "Quasi-invariant measure",
    "same_as": {
      "wikidata": "Q7269454",
      "wikipedia": "Quasi-invariant measure"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Quasi-invariant_measure"
      }
    ],
    "tag": "FCNQP0A60D"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/FCNQP0A60D/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/FCNQP0A60D.md",
  "msc": [
    "28C10",
    "28D05",
    "22D40"
  ],
  "name": "Quasi-invariant measure",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/FCN/FCNQP0A60D.md",
  "status": null,
  "tag": "FCNQP0A60D",
  "uri": "m2://FCNQP0A60D",
  "url": "https://math2.tamnd.com/wiki/quasi-invariant-measure/",
  "writing": {
    "tag": "FCNQP0A60D",
    "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
      }
    ]
  }
}