{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "64F9VMG27D"
    }
  ],
  "body": "\n## Summary\n\nAn invariant mean on a [[group|G5S1M9]] is a way to average all bounded functions on the group.\nThe average does not change when the function is translated.\nA discrete group has a left-invariant mean exactly when it is [[amenable|CEAT9BESKK]].\nInvariant means are the functional form of finitely additive invariant probability measures.\n\n## Explanations\n\n### L1 human\n\nOn a finite group one averages a function by adding its values and dividing by the number of elements.\nThis average does not change when the function is shifted by a group element.\nFor an infinite group such as the integers, one would like an average of every bounded sequence that ignores shifts.\nSuch averages exist for the integers, but they cannot be written down explicitly.\nFor the free group on two generators, no such average exists at all.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "kind": "concept",
    "msc": [
      "43A07",
      "28C10"
    ],
    "name": "Invariant mean",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Amenable_group"
      }
    ],
    "tag": "ZYNY2VFECQ"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/ZYNY2VFECQ/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/ZYNY2VFECQ.md",
  "msc": [
    "43A07",
    "28C10"
  ],
  "name": "Invariant mean",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/ZYN/ZYNY2VFECQ.md",
  "status": null,
  "tag": "ZYNY2VFECQ",
  "uri": "m2://ZYNY2VFECQ",
  "url": "https://math2.tamnd.com/wiki/invariant-mean/",
  "writing": {
    "tag": "ZYNY2VFECQ",
    "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
      }
    ]
  }
}