{
  "id": "1202.5792",
  "version": "v2",
  "published": "2012-02-26T21:54:43.000Z",
  "updated": "2012-07-15T18:58:48.000Z",
  "title": "Relations between $\\mathcal{L}^p$- and Pointwise Convergence of Families of Functions Indexed by the Unit Interval",
  "authors": [
    "Vaios Laschos",
    "Christian Mönch"
  ],
  "comment": "13 pages",
  "categories": [
    "math.CA"
  ],
  "abstract": "We construct a variety of mappings of the unit interval into $\\mathcal{L}^p([0,1])$ to generalize classical examples of $\\mathcal{L}^p$-convergence of sequences of functions with simultaneous pointwise divergence. By establishing relations between the regularity of the functions in the image of the mappings and the topology of $[0,1]$, we obtain examples which are $\\mathcal{L}^p$-continuous but exhibit discontinuity in a pointwise sense to different degrees. We conclude by proving an Egorov-type theorem, namely that if almost every function in the image is continuous, then we can remove a set of arbitrarily small measure from the index set $[0,1]$ and establish pointwise limits for all functions in the remaining image.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-07-15T18:58:48.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "unit interval",
      "pointwise convergence",
      "egorov-type theorem",
      "arbitrarily small measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}