Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

A classification of Baire-1 functions
HTML articles powered by AMS MathViewer

by P. Kiriakouli PDF
Trans. Amer. Math. Soc. 351 (1999), 4599-4609 Request permission

Abstract:

In this paper we give some topological characterizations of bounded Baire-1 functions using some ranks. Kechris and Louveau classified the Baire-1 functions to the subclasses $\mathbb {B}^\xi _1(K)$ for every $\xi <\omega _1$ (where $K$ is a compact metric space). The first basic result of this paper is that for $\xi <\omega$, $f\in \mathbb {B}^{\xi +1}_1(K)$ iff there exists a sequence $(f_n)$ of differences of bounded semicontinuous functions on $K$ with $f_n\to f$ pointwise and $\gamma ((f_n))\le \omega ^\xi$ (where “$\gamma$” denotes the convergence rank). This extends the work of Kechris and Louveau who obtained this result for $\xi =1$. We also show that the result fails for $\xi \ge \omega$. The second basic result of the paper involves the introduction of a new ordinal-rank on sequences $(f_n)$, called the $\delta$-rank, which is smaller than the convergence rank $\gamma$. This result yields the following characterization of $\mathbb {B}^\xi _1(K): f\in \mathbb {B}^\xi _1(K)$ iff there exists a sequence $(f_n)$ of continuous functions with $f_n\to f$ pointwise and $\delta ((f_n))\le \omega ^{\xi -1}$ if $1\le \xi <\omega$, resp. $\delta ((f_n))\le \omega ^\xi$ if $\xi \ge \omega$.
References
  • D. C. Gillespie and H. A. Hurwicz, On sequences of continuous functions having continuous limits, Trans. Amer. Math. Soc. 32 (1930), 527–543.
  • R. Haydon, E. Odell and H. P. Rosenthal, Certain subclasses of Baire-$1$ functions with Banach space applications, Longhorn Notes, University of Texas at Austin Functional Analysis Seminar 1987–89.
  • A. S. Kechris and A. Louveau, A classification of Baire class $1$ functions, Trans. Amer. Math. Soc. 318 (1990), no. 1, 209–236. MR 946424, DOI 10.1090/S0002-9947-1990-0946424-3
  • P. Kiriakouli and S. Negrepontis, A classification of Baire-$1$ functions, unpublished.
  • Z. Zalcwasser, Sur une propriete du champ des fonctions continues, Studia Math. 2 (1930), 63–67.
Similar Articles
Additional Information
  • P. Kiriakouli
  • Affiliation: Department of Mathematics, University of Athens, Panepistimiopolis 15784, Athens, Greece
  • Received by editor(s): July 11, 1994
  • Received by editor(s) in revised form: December 28, 1995
  • Published electronically: July 21, 1999
  • © Copyright 1999 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 351 (1999), 4599-4609
  • MSC (1991): Primary 03E15, 04A15, 46B99, 54C50
  • DOI: https://doi.org/10.1090/S0002-9947-99-01907-8
  • MathSciNet review: 1407705