{
  "id": "1208.3187",
  "version": "v2",
  "published": "2012-08-15T19:37:53.000Z",
  "updated": "2013-05-15T15:11:53.000Z",
  "title": "On the Law of Large Numbers for Nonmeasurable Identically Distributed Random Variables",
  "authors": [
    "Alexander R. Pruss"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $\\Omega$ be a countable infinite product $\\Omega^\\N$ of copies of the same probability space $\\Omega_1$, and let ${\\Xi_n}$ be the sequence of the coordinate projection functions from $\\Omega$ to $\\Omega_1$. Let $\\Psi$ be a possibly nonmeasurable function from $\\Omega_1$ to $\\R$, and let $X_n(\\omega) = \\Psi(\\Xi_n(\\omega))$. Then we can think of ${X_n}$ as a sequence of independent but possibly nonmeasurable random variables on $\\Omega$. Let $S_n = X_1+...+X_n$. By the ordinary Strong Law of Large Numbers, we almost surely have $E_*[X_1] \\le \\liminf S_n/n \\le \\limsup S_n/n \\le E^*[X_1]$, where $E_*$ and $E^*$ are the lower and upper expectations. We ask if anything more precise can be said about the limit points of $S_n/n$ in the non-trivial case where $E_*[X_1] < E^*[X_1]$, and obtain several negative answers. For instance, the set of points of $\\Omega$ where $S_n/n$ converges is maximally nonmeasurable: it has inner measure zero and outer measure one.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-05-15T15:11:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F10",
      "60F05",
      "28A12"
    ],
    "keywords": [
      "nonmeasurable identically distributed random variables",
      "large numbers",
      "inner measure zero",
      "coordinate projection functions",
      "ordinary strong law"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1208.3187P"
    }
  }
}