Extension of functions with small oscillation

Abstract

A classical theorem of Kuratowski says that every Baire one function on a Gsubspace of a Polish (= separable completely metriz- able) 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 �(f). We prove a re- finement of Kuratowski's theorem: if Y is a subspace of a metric space X and f is a real-valued function on Y such thatY (f) < ! � , � < !1, then f has an extension F onto X so thatX(F)� ! � . We also show that if f is a continuous real valued function on Y, then f has an exten- sion F onto X so thatX (F) � 3. An example is constructed to show that this result is optimal.