Extension of functions with small oscillation

Denny H. Leung¹ ; Wee-Kee Tang²

¹ Department of Mathematics National University of Singapore 2 Science Drive 2 Singapore 117543
² Mathematics and Mathematics Education National Institute of Education Nanyang Technological University 1 Nanyang Walk Singapore 637616

Fundamenta Mathematicae, Tome 192 (2006) no. 2, pp. 183-193.

Résumé

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.

Détail

DOI : 10.4064/fm192-2-6

Keywords: classical theorem kuratowski says every baire function delta subspace polish separable completely metrizable space extended baire function kechris louveau introduced finer gradation baire functions small baire classes baire function assigned class hierarchy depending its oscillation index beta prove refinement kuratowskis theorem subspace metric space real valued function beta omega alpha alpha omega has extension beta leq omega alpha continuous real valued function has extension beta leq example constructed result optimal