{
  "id": "1909.03505",
  "version": "v1",
  "published": "2019-09-08T16:50:58.000Z",
  "updated": "2019-09-08T16:50:58.000Z",
  "title": "Differentiation of measures on a non-separable space, and the Radon-Nikodym theorem",
  "authors": [
    "Oleksii Mostovyi",
    "Pietro Siorpaes"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "Given positive measures $\\nu,\\mu$ on an arbitrary measurable space $(\\Omega, \\mathcal F)$, we construct a sequence of finite partitions $(\\pi_n)_n$ of $(\\Omega, \\mathcal F)$ s.t. $$ \\sum_{A\\in \\pi_n: \\mu(A)>0} 1_{A} \\frac{\\nu(A)}{\\mu(A)} \\longrightarrow \\frac{d\\nu^a}{d\\mu} \\quad \\mu \\text{ a.e. as } n\\to \\infty . $$ As an application, we modify the probabilistic proof of the Radon-Nikodym Theorem so that it uses convergence along a properly chosen sequence (instead of along a net), and so that it does not rely on the martingale convergence theorem (nor any probability theory), obtaining a completely elementary proof.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-09-08T16:50:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A15",
      "28A25"
    ],
    "keywords": [
      "radon-nikodym theorem",
      "non-separable space",
      "differentiation",
      "martingale convergence theorem",
      "probability theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}