Functions of Baire class one

Denny H. Leung; Wee-Kee Tang

Fundamenta Mathematicae (2003)

  • Volume: 179, Issue: 3, page 225-247
  • ISSN: 0016-2736

Abstract

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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 ξ · ω ξ 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 s u p β ( f ) ω ξ and γ ( ( f ) ) ω ξ . 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 ξ and β ( g ) ω ξ , then β ( f g ) ω ξ , where ξ = maxξ₁+ξ₂,ξ₂+ξ₁. These results do not assume the boundedness of the functions involved.

How to cite

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Denny H. Leung, and Wee-Kee Tang. "Functions of Baire class one." Fundamenta Mathematicae 179.3 (2003): 225-247. <http://eudml.org/doc/283074>.

@article{DennyH2003,
abstract = {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 = maxξ₁+ξ₂,ξ₂+ξ₁. These results do not assume the boundedness of the functions involved.},
author = {Denny H. Leung, Wee-Kee Tang},
journal = {Fundamenta Mathematicae},
keywords = {Baire-1 functions; convergence index; oscillation index},
language = {eng},
number = {3},
pages = {225-247},
title = {Functions of Baire class one},
url = {http://eudml.org/doc/283074},
volume = {179},
year = {2003},
}

TY - JOUR
AU - Denny H. Leung
AU - Wee-Kee Tang
TI - Functions of Baire class one
JO - Fundamenta Mathematicae
PY - 2003
VL - 179
IS - 3
SP - 225
EP - 247
AB - 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 = maxξ₁+ξ₂,ξ₂+ξ₁. These results do not assume the boundedness of the functions involved.
LA - eng
KW - Baire-1 functions; convergence index; oscillation index
UR - http://eudml.org/doc/283074
ER -

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