{
  "id": "math/0005013",
  "version": "v1",
  "published": "2000-05-02T06:13:21.000Z",
  "updated": "2000-05-02T06:13:21.000Z",
  "title": "Functions of Baire class one",
  "authors": [
    "Denny H. Leung",
    "Wee-Kee Tang"
  ],
  "categories": [
    "math.CA",
    "math.FA"
  ],
  "abstract": "Let $K$ be a compact metric space. A real-valued function on $K$ is said to be of Baire class one (Baire-1) if it is the pointwise limit of a sequence of continuous functions. In this paper, we study two well known ordinal indices of Baire-1 functions, the oscillation index $\\beta$ and the convergence index $\\gamma$. It is shown that these two indices are fully compatible in the following sense : a Baire-1 function $f$ satisfies $\\beta(f) \\leq \\omega^{\\xi_1} \\cdot \\omega^{\\xi_2}$ for some countable ordinals $\\xi_1$ and $\\xi_2$ if and only if there exists a sequence of Baire-1 functions $(f_n)$ converging to $f$ pointwise such that $\\sup_n\\beta(f_n) \\leq \\omega^{\\xi_1}$ and $\\gamma((f_n)) \\leq \\omega^{\\xi_2}$. We also obtain an extension result for Baire-1 functions analogous to the Tietze Extension Theorem. Finally, it is shown that if $\\beta(f) \\leq \\omega^{\\xi_1}$ and $\\beta(g) \\leq \\omega^{\\xi_2},$ then $\\beta(fg) \\leq \\omega^{\\xi},$ where $\\xi=\\max\\{\\xi_1+\\xi_2, \\xi_2+\\xi_1}\\}.$ These results do not assume the boundedness of the functions involved.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2000-05-02T06:13:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "26A21",
      "03E15",
      "54C30"
    ],
    "keywords": [
      "baire class",
      "compact metric space",
      "tietze extension theorem",
      "extension result",
      "convergence index"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2000math......5013L"
    }
  }
}