Functions of Baire class one

Denny H. Leung¹ ; Wee-Kee Tang²

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

Fundamenta Mathematicae, Tome 179 (2003) no. 3, pp. 225-247.

Résumé

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.

Détail

DOI : 10.4064/fm179-3-3

Keywords: compact metric space real valued function said baire class baire pointwise limit sequence continuous functions study known ordinal indices baire functions oscillation index beta convergence index gamma shown these indices fully compatible following sense baire function satisfies beta leq omega cdot omega countable ordinals only there exists sequence baire functions converging pointwise sup beta leq omega gamma leq omega obtain extension result baire functions analogous tietze extension theorem finally shown beta leq omega beta leq omega beta leq omega where max these results assume boundedness functions involved