{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Problem\n\nLet $(m_n)_{n \\ge 0}$ be a [[sequence|0G9EKM0E15]] of real numbers.\nDecide whether there is a positive Borel [[measure|PR11BZQQPG]] $\\mu$ on the closed interval $[0, 1]$ such that\n$$m_n = \\int_{[0,1]} x^n \\, d\\mu(x) \\quad \\text{for all } n \\ge 0.$$\nThe numbers $m_n$ are then the [[moments|01XAHPADBA]] of $\\mu$.\n\n## Remarks\n\nThe problem is solved.\nWrite $(\\Delta m)_n = m_{n+1} - m_n$ for the difference operator.\nHausdorff proved that such a measure exists if and only if the sequence is completely monotone, that is,\n$$(-1)^k (\\Delta^k m)_n \\ge 0 \\quad \\text{for all } n, k \\ge 0.$$\nWhen the measure exists, it is unique.\nUniqueness holds because polynomials are dense in the continuous functions on $[0, 1]$ by the Weierstrass approximation theorem.\nFor example, $m_n = 1/(n + 1)$ is the moment sequence of Lebesgue measure on $[0, 1]$.\n\n## Explanations\n\n### L1 human\n\nThe moments of a mass distribution on $[0, 1]$ are its total mass, its mean position, its mean squared position, and so on.\nThe problem asks which lists of numbers can occur as such moments.\nNot every list works: for instance, the moments can never increase, since $x^{n+1} \\le x^n$ on $[0, 1]$.\nHausdorff found an exact test, a sign condition on all repeated differences of the list, and showed that the list then determines the distribution.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "about": [
      "01XAHPADBA",
      "PR11BZQQPG"
    ],
    "history": {
      "first_stated": "1921",
      "posed_by": "Felix Hausdorff"
    },
    "kind": "problem",
    "msc": [
      "44A60",
      "28A12"
    ],
    "name": "Hausdorff moment problem",
    "same_as": {
      "wikidata": "Q5682821",
      "wikipedia": "Hausdorff moment problem"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Hausdorff_moment_problem"
      }
    ],
    "tag": "G7VG61YD4J",
    "uses": [
      "PR11BZQQPG",
      "0G9EKM0E15",
      "01XAHPADBA"
    ]
  },
  "kind": "problem",
  "link": "https://math2.tamnd.com/node/G7VG61YD4J/",
  "links": [
    {
      "field": "about",
      "target": "01XAHPADBA"
    },
    {
      "field": "about",
      "target": "PR11BZQQPG"
    },
    {
      "field": "uses",
      "target": "PR11BZQQPG"
    },
    {
      "field": "uses",
      "target": "0G9EKM0E15"
    },
    {
      "field": "uses",
      "target": "01XAHPADBA"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/G7VG61YD4J.md",
  "msc": [
    "44A60",
    "28A12"
  ],
  "name": "Hausdorff moment problem",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/G7V/G7VG61YD4J.md",
  "status": null,
  "tag": "G7VG61YD4J",
  "uri": "m2://G7VG61YD4J",
  "url": "https://math2.tamnd.com/wiki/hausdorff-moment-problem/",
  "writing": {
    "tag": "G7VG61YD4J",
    "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
      }
    ]
  }
}