Functions of Baire class one

Volume 179 / 2003

Denny H. Leung, Wee-Kee Tang Fundamenta Mathematicae 179 (2003), 225-247 MSC: Primary 26A21; Secondary 03E15, 54C30. DOI: 10.4064/fm179-3-3


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. 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 $(f_{n})$ of Baire-$1$ functions 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.


  • Denny H. LeungDenny H. Leung
    Department of Mathematics
    National University of Singapore
    2 Science Drive 2, Singapore 117543
  • Wee-Kee TangWee-Kee Tang
    Mathematics and Mathematics Education
    National Institute of Education
    Nanyang Technological University
    1 Nanyang Walk, Singapore 637616

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