{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $(X, \\mathcal{A})$ be a measurable space and let $T \\colon X \\to X$ be a [[measurable map|YYNCJSR0PJ]].\nA [[measure|PR11BZQQPG]] $\\mu$ on $(X, \\mathcal{A})$ is *$T$-invariant* if\n$$\\mu(T^{-1}(A)) = \\mu(A) \\quad \\text{for every } A \\in \\mathcal{A}.$$\nLet a family of measurable maps be given, such as a flow $(\\varphi_t)$ or a group action.\nThen $\\mu$ is *invariant* if it is invariant under each map of the family.\n\n## Remarks\n\nEquivalently, the [[pushforward measure|3N66TM3TSQ]] $T_* \\mu$ equals $\\mu$, or $T$ is a [[measure-preserving transformation|556WBDAHNP]] of $(X, \\mathcal{A}, \\mu)$ to itself.\nFor an integrable function $g$, invariance gives $\\int g \\circ T \\, d\\mu = \\int g \\, d\\mu$.\nThe invariant probability measures of $T$ form a convex set, whose extreme points are the ergodic ones.\n\nExample: the doubling map $T(x) = 2x \\bmod 1$ on $[0, 1)$ and Lebesgue measure $\\lambda$.\nFor an interval $[a, b) \\subseteq [0, 1)$, one has $T^{-1}([a, b)) = [a/2, b/2) \\cup [(a+1)/2, (b+1)/2)$, of length $b - a$.\nSince intervals generate the Borel sets and are closed under intersection, $\\lambda$ is $T$-invariant.\n\nNon-example: the map $T(x) = x^2$ on $[0, 1]$ with Lebesgue measure $\\lambda$.\nOne has $T^{-1}([0, 1/4]) = [0, 1/2]$, so $\\lambda(T^{-1}([0, 1/4])) = 1/2 \\ne 1/4$, and $\\lambda$ is not $T$-invariant.\nThe point masses at $0$ and at $1$ are $T$-invariant, since $0$ and $1$ are fixed points.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "B232EZ1FMB",
    "kind": "definition",
    "msc": [
      "37A05",
      "28D05",
      "37C40"
    ],
    "name": "Invariant measure",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Invariant_measure"
      }
    ],
    "tag": "HX6SS7WYQV",
    "uses": [
      "PR11BZQQPG",
      "YYNCJSR0PJ",
      "556WBDAHNP",
      "3N66TM3TSQ"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/HX6SS7WYQV/",
  "links": [
    {
      "field": "defines",
      "target": "B232EZ1FMB"
    },
    {
      "field": "uses",
      "target": "PR11BZQQPG"
    },
    {
      "field": "uses",
      "target": "YYNCJSR0PJ"
    },
    {
      "field": "uses",
      "target": "556WBDAHNP"
    },
    {
      "field": "uses",
      "target": "3N66TM3TSQ"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/HX6SS7WYQV.md",
  "msc": [
    "37A05",
    "28D05",
    "37C40"
  ],
  "name": "Invariant measure",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/HX6/HX6SS7WYQV.md",
  "status": null,
  "tag": "HX6SS7WYQV",
  "uri": "m2://HX6SS7WYQV",
  "url": "https://tamnd.github.io/math2/wiki/invariant-measure-definition/",
  "writing": {
    "tag": "HX6SS7WYQV",
    "kind": "definition",
    "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
      }
    ]
  }
}