Extension of Functions with Small Oscillation

Abstract

A classical theorem of Kuratowski says that every Baire one function on a G_δsubspace 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 β(f). We prove a refinement of Kuratowski&#39;s theorem: if Y is a subspace of a metric space X and f is a real-valued function on Y such that β_{Y}(f)<ω^α, α< ω_1, then f has an extension F onto X so that β_X(F)is not more than ω^α. We also show that if f is a continuous real valued function on Y, then f has an extension F onto X so that β_{X}(F)is not more than 3. An example is constructed to show that this result is optimal.