Extension of Functions with Small Oscillation
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 heirarchy 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 realvalued function on Y such that \beta_{Y}(f)<\omega^{\alpha}, \alpha < \omega_1, then f has an extension F onto X so that \beta_X(F)is not more than \omega^{\alpha}. We also show that if f is a continuous real valued function on Y, then f has an extension F onto X so that \beta_{X}(F)is not more than 3. An example is constructed to show that this result is optimal.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 May 2005
 DOI:
 10.48550/arXiv.math/0505168
 arXiv:
 arXiv:math/0505168
 Bibcode:
 2005math......5168L
 Keywords:

 Classical Analysis and ODEs;
 Functional Analysis;
 26A21;
 03E15;
 54C30