A classical theorem of Kuratowski says that every Baire one function on a 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 β(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 , α < ω₁, then f has an extension F to X so that . We also show that if f is a continuous real-valued function on Y, then f has an extension F to X so that An example is constructed to show that this result is optimal.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm192-2-6,
author = {Denny H. Leung and Wee-Kee Tang},
title = {Extension of functions with small oscillation},
journal = {Fundamenta Mathematicae},
volume = {189},
year = {2006},
pages = {183-193},
zbl = {1112.26003},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm192-2-6}
}
Denny H. Leung; Wee-Kee Tang. Extension of functions with small oscillation. Fundamenta Mathematicae, Tome 189 (2006) pp. 183-193. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm192-2-6/