Extension of functions with small oscillation

• Autorzy: Denny H. Leung , Wee-Kee Tang
• Języki publikacji: EN

Abstrakty

EN

A classical theorem of Kuratowski says that every Baire one function on a $G_{δ}$ 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 $β_{Y}(f) < ω^{α}$, α < ω₁, then f has an extension F to X so that $β_{X}(F) ≤ ω^{α}$. We also show that if f is a continuous real-valued function on Y, then f has an extension F to X so that $β_{X}(F) ≤ 3.$ An example is constructed to show that this result is optimal.

Kategorie tematyczne

• 26A21: Classification of real functions; Baire classification of sets and functions
• 03E15: Descriptive set theory
• 54C30: Real-valued functions

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2006
• Tom: 192
• Numer: 2
• Strony: 183-193

Daty

wydano

2006

Twórcy

autor

Denny H. Leung

• Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543

autor

Wee-Kee Tang

• Mathematics and Mathematics Education, National Institute of Education, Nanyang Technological University, 1 Nanyang Walk, Singapore 637616

Identyfikatory

DOI

10.4064/fm192-2-6