Functions of Baire class one

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

Abstrakty

EN

Let K be a compact metric space. A real-valued function on K is said to be of Baire class one (Baire-1) if it is the pointwise limit of a sequence of continuous functions. We study two well known ordinal indices of Baire-1 functions, the oscillation index β and the convergence index γ. It is shown that these two indices are fully compatible in the following sense: a Baire-1 function f satisfies $β(f) ≤ ω^{ξ₁} · ω^{ξ₂}$ for some countable ordinals ξ₁ and ξ₂ if and only if there exists a sequence (fₙ) of Baire-1 functions converging to f pointwise such that $supₙβ(fₙ) ≤ ω^{ξ₁}$ and $γ((fₙ)) ≤ ω^{ξ₂}$. We also obtain an extension result for Baire-1 functions analogous to the Tietze Extension Theorem. Finally, it is shown that if $β(f) ≤ ω^{ξ₁}$ and $β(g) ≤ ω^{ξ₂}$, then $β(fg) ≤ ω^{ξ}$, where ξ = max{ξ₁+ξ₂,ξ₂+ξ₁}. These results do not assume the boundedness of the functions involved.

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: 2003
• Tom: 179
• Numer: 3
• Strony: 225-247

Daty

wydano

2003

Twórcy

autor

Denny H. Leung

• Denny H. Leung, Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543

autor

Wee-Kee Tang

• Wee-Kee Tang, Mathematics and Mathematics Education, National Institute of Education, Nanyang Technological University, 1 Nanyang Walk, Singapore 637616

Identyfikatory

DOI

10.4064/fm179-3-3