{
  "id": "1202.2106",
  "version": "v2",
  "published": "2012-02-09T20:35:36.000Z",
  "updated": "2012-05-14T08:21:25.000Z",
  "title": "A Simple Proof of Vitali's Theorem for Signed Measures",
  "authors": [
    "Tony Samuel"
  ],
  "journal": "The American Mathematical Monthly. Volume 120. Number 7. pp. 654-659. 2013",
  "categories": [
    "math.DS",
    "math.MG"
  ],
  "abstract": "There are several theorems named after the Italian mathematician Vitali. In this note we provide a simple proof of an extension of Vitali's Theorem on the existence of non-measurable sets. Specifically, we show, without using any decomposition theorems, that there does not exist a non-trivial, atom-less, $\\sigma$-additive and translation invariant set function $\\mathcal{L}$ from the power set of the real line to the extended real numbers with $\\mathcal{L}([0,1]) = 1$. (Note that $\\mathcal{L}$ is not assumed to be non-negative.)",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-05-14T08:21:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28Axx"
    ],
    "keywords": [
      "simple proof",
      "vitalis theorem",
      "signed measures",
      "translation invariant set function",
      "italian mathematician vitali"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1202.2106S"
    }
  }
}