{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Problem\n\nLet $n \\ge 1$.\nIs there a function $\\mu$, defined on all subsets of $\\R^n$, with values in $[0, \\infty]$, such that:\n\n1. $\\mu$ is finitely additive: $\\mu(A \\cup B) = \\mu(A) + \\mu(B)$ for disjoint $A, B \\subseteq \\R^n$.\n2. $\\mu$ is invariant under isometries: $\\mu(g(A)) = \\mu(A)$ for every isometry $g$ of $\\R^n$, such as a [[translation|087VJM5Q77]] or a [[rotation|0HFREYRSZP]].\n3. $\\mu([0, 1]^n) = 1$.\n\n## Remarks\n\nSuch a function is a [[finitely additive measure|KFWQ24D9M6]] on the power set.\nThe countably additive version has a negative answer, by the Vitali set.\nHausdorff showed with his paradox on the sphere that the answer is negative for $n \\ge 3$.\nBanach showed in 1923 that the answer is positive for $n = 1$ and $n = 2$; such a function can even be chosen to agree with [[Lebesgue measure|2Z2QVWZ262]] on Lebesgue measurable sets.\nThese functions are called Banach measures, and they are not unique.\nThe difference between the dimensions is explained by amenability.\nThe isometry groups of $\\R$ and $\\R^2$ are amenable, and the rotation group of $\\R^3$ contains a free group on two generators.\n\n## Explanations\n\n### L1 human\n\nThe question asks whether every set of points, however wild, can be given a length, an area or a volume.\nThe size should add up for two pieces and should not change when a set is moved rigidly.\nOn the line and in the plane this is possible, but in space of dimension three or more it is not.\nIn space, the Banach–Tarski paradox even cuts a ball into finitely many pieces that can be reassembled into two balls of the same size.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "about": [
      "2Z2QVWZ262"
    ],
    "history": {
      "first_stated": "1914",
      "posed_by": "Felix Hausdorff"
    },
    "kind": "problem",
    "msc": [
      "28C10",
      "43A07",
      "28A12"
    ],
    "name": "Banach measure problem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Banach_measure"
      }
    ],
    "tag": "JXK6NRDH4V",
    "uses": [
      "2Z2QVWZ262",
      "KFWQ24D9M6",
      "0HFREYRSZP",
      "087VJM5Q77"
    ]
  },
  "kind": "problem",
  "link": "https://math2.tamnd.com/node/JXK6NRDH4V/",
  "links": [
    {
      "field": "about",
      "target": "2Z2QVWZ262"
    },
    {
      "field": "uses",
      "target": "2Z2QVWZ262"
    },
    {
      "field": "uses",
      "target": "KFWQ24D9M6"
    },
    {
      "field": "uses",
      "target": "0HFREYRSZP"
    },
    {
      "field": "uses",
      "target": "087VJM5Q77"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/JXK6NRDH4V.md",
  "msc": [
    "28C10",
    "43A07",
    "28A12"
  ],
  "name": "Banach measure problem",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/JXK/JXK6NRDH4V.md",
  "status": null,
  "tag": "JXK6NRDH4V",
  "uri": "m2://JXK6NRDH4V",
  "url": "https://math2.tamnd.com/wiki/banach-measure-problem/",
  "writing": {
    "tag": "JXK6NRDH4V",
    "kind": "problem",
    "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
      }
    ]
  }
}