{
  "id": "1408.2097",
  "version": "v2",
  "published": "2014-08-09T13:43:47.000Z",
  "updated": "2015-02-26T18:58:35.000Z",
  "title": "On continuous Polish group actions and equivalence relations",
  "authors": [
    "Nikolaos E. Sofronidis"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "Let $X = \\left\\{ P \\in [0,1]^{\\bf N} : \\left( \\forall \\nu \\in {\\bf N} \\right) \\left( P \\left( \\{ \\nu \\} \\right) > 0 \\right) \\wedge \\sum\\limits_{ \\nu = 0 }^{ \\infty } P \\left( \\{ \\nu \\} \\right) = 1 \\right\\} $ be the Polish space of probability measures on ${\\bf N}$, each of which assigns positive probability to every elementary event, while for any $P \\in X$, let ${\\Gamma }_{P} = \\left\\{ \\xi \\in L^{1}( {\\bf N} , P ) : \\left( \\forall \\nu \\in {\\bf N} \\right) \\left( \\xi ( \\nu ) > 0 \\right) \\wedge \\sum\\limits_{ \\nu = 0 }^{ \\infty } \\xi ( \\nu ) P \\left( \\{ \\nu \\} \\right) = 1 \\right\\} $ and let ${\\Phi }_{P} : {\\Gamma }_{P} \\ni \\xi \\mapsto {\\Phi }_{P}( \\xi ) \\in X$ be defined by the relation $\\left( {\\Phi }_{P}( \\xi ) \\right) \\left( \\{ \\nu \\} \\right) = \\xi ( \\nu ) P \\left( \\{ \\nu \\} \\right) $, whenever $\\nu \\in {\\bf N}$. If we consider the equivalence relation $E = \\left\\{ (P,Q) \\in X^{2} : \\left( \\exists \\xi \\in {\\Gamma }_{P} \\right) \\left( Q = {\\Phi }_{P}( \\xi ) \\right) \\right\\} $, the Polish space ${\\bf P} = \\left\\{ {\\bf x} \\in {\\ell }^{1} \\left( {\\bf R} \\right) : \\left( \\forall n \\in {\\bf N} \\right) \\left( {\\bf x}(n) > 0 \\right) \\right\\} $ and the commutative Polish group ${\\bf G} = \\left\\{ {\\bf g} \\in (0 , \\infty )^{\\bf N} : \\lim\\limits_{n \\rightarrow \\infty }{\\bf g}(n) = 1 \\right\\} $, while we set $\\left( {\\bf g} \\cdot {\\bf x} \\right) (n) = {\\bf g}(n){\\bf x}(n)$, whenever ${\\bf g} \\in {\\bf G}$, ${\\bf x} \\in {\\bf P}$ and $n \\in {\\bf N}$, then $E$ is definable and it admits a strong approximation by the turbulent Polish group action of ${\\bf G}$ on ${\\bf P}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-08-09T13:43:47.000Z",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-02-26T18:58:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03E15",
      "22A99",
      "40A99"
    ],
    "keywords": [
      "continuous polish group actions",
      "equivalence relation",
      "turbulent polish group action",
      "polish space",
      "assigns positive probability"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1408.2097S"
    }
  }
}