{
  "id": "2006.07372",
  "version": "v1",
  "published": "2020-06-12T16:41:25.000Z",
  "updated": "2020-06-12T16:41:25.000Z",
  "title": "Sensitive Random Variables are Dense in Every $L^{p}(\\mathbb{R}, \\mathscr{B}_{\\mathbb{R}}, \\mathbb{P})$",
  "authors": [
    "Yu-Lin Chou"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We show that, for every $1 \\leq p < +\\infty$ and for every Borel probability measure $\\mathbb{P}$ over $\\mathbb{R}$, every element of $L^{p}(\\mathbb{R}, \\mathscr{B}_{\\mathbb{R}}, \\mathbb{P})$ is the $L^{p}$-limit of some sequence of random variables in $L^{p}(\\mathbb{R}, \\mathscr{B}_{\\mathbb{R}}, \\mathbb{P})$ that are Lebesgue-almost everywhere differentiable with derivatives having norm greater than any pre-specified real number at every point of differentiability. In general, this result provides, in some direction, a finer description of an $L^{p}$-approximation for $L^{p}$ functions on $\\mathbb{R}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-12T16:41:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60A10",
      "46E30",
      "26A24"
    ],
    "keywords": [
      "sensitive random variables",
      "borel probability measure",
      "norm greater",
      "pre-specified real number",
      "finer description"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}