Extension of Functions with Small Oscillation

2005/05/10 by Denny H. Leung, Leung, Denny H., Wee-Kee Tang +1
Mathematics · #03E15 #26A21 #54C30 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #math.CA #math.FA #msc:03E15 #msc:26A21 #msc:54C30

paper · pdf · doi:10.48550/arxiv.math/0505168

arxiv created 2005/05/10 · arxiv updated 2009/12/01

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'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< ω₁, 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.

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