Extension of functions with small oscillation

Denny H. Leung, and Wee-Kee Tang. "Extension of functions with small oscillation." Fundamenta Mathematicae 192.2 (2006): 183-193. <http://eudml.org/doc/282687>.

@article{DennyH2006,
abstract = {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.},
author = {Denny H. Leung, Wee-Kee Tang},
journal = {Fundamenta Mathematicae},
keywords = {Baire-1 function; oscillation rank},
language = {eng},
number = {2},
pages = {183-193},
title = {Extension of functions with small oscillation},
url = {http://eudml.org/doc/282687},
volume = {192},
year = {2006},
}

TY - JOUR
AU - Denny H. Leung
AU - Wee-Kee Tang
TI - Extension of functions with small oscillation
JO - Fundamenta Mathematicae
PY - 2006
VL - 192
IS - 2
SP - 183
EP - 193
AB - 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.
LA - eng
KW - Baire-1 function; oscillation rank
UR - http://eudml.org/doc/282687
ER -