Extension of functions with small oscillation
Volume 192 / 2006
Abstract
A classical theorem of Kuratowski says that every Baire one function on a $G_{\delta }$ subspace of a Polish ($=$ separable completely metrizable) space $X$ can be extended to a Baire one function on $X$. Kechris and Louveau introduced a finer gradation of Baire one functions into small Baire classes. A Baire one function $f$ is assigned into a class in this hierarchy depending on its oscillation index $\beta (f)$. We prove a refinement of Kuratowski's theorem: if $Y$ is a subspace of a metric space $X$ and $f$ is a real-valued function on $Y$ such that $\beta _{Y}(f)<\omega ^{\alpha }$, $\alpha <\omega _{1}$, then $f$ has an extension $F$ to $X$ so that $\beta _{X}(F)\leq \omega ^{\alpha }$. We also show that if $f$ is a continuous real-valued function on $Y,$ then $f$ has an extension $F$ to $X$ so that $\beta _{X}(F) \leq 3.$ An example is constructed to show that this result is optimal.
