{
  "id": "1807.05624",
  "version": "v1",
  "published": "2018-07-15T22:13:48.000Z",
  "updated": "2018-07-15T22:13:48.000Z",
  "title": "The conjecture of Ulam on the invariance of measure on Hilbert cube",
  "authors": [
    "Soon-Mo Jung"
  ],
  "comment": "14 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "A conjecture of Ulam states that the standard probability measure $\\pi$ on the Hilbert cube $I^\\omega$ is invariant under the induced metric $d_a$ provided the sequence $a = \\{ a_i \\}$ of positive numbers satisfies the condition $\\sum\\limits_{i=1}^\\infty a_i^2 < \\infty$. In this paper, we prove this conjecture in the affirmative. More precisely, we prove that if there exists a surjective $d_a$-isometry $f : E_1 \\to E_2$, where $E_1$ and $E_2$ are Borel subsets of $I^\\omega$, then $\\pi(E_1) = \\pi(E_2)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-07-15T22:13:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28C10",
      "28A35",
      "28A12",
      "28A75"
    ],
    "keywords": [
      "hilbert cube",
      "conjecture",
      "invariance",
      "standard probability measure",
      "ulam states"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}