Functions of Baire class one
Volume 179 / 2003
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. 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.
