Baire第一类函数

摘要：

设$K$是一个紧度量空间. $K$上的实值函数称为Baire类1（Baire-1），如果它是连续函数序列的逐点极限。本文研究了Baire-1函数的两个著名的序指标：振荡指标$β$和收敛指标$γ$。证明了这两个指标在以下意义下是完全相容的：对于可数序数$ξ_1$和$ξ_2$，Baire-1函数$f$满足$β(f) \leq ω^{ξ_1} \cdot ω^{ξ_2}$当且仅当存在一列逐点收敛到$f$的Baire-1函数$(f_n)$，使得$\sup_nβ(f_n) \leq ω^{ξ_1}$和$γ((f_n)) \leq ω^{ξ_2}$ .我们还得到了类似于Tietze扩张定理的Baire-1函数的一个扩张结果。最后，它表明，如果$β(f) \leq ω^{ξ_1}$和$β(g) \leq ω^{ξ_2},$，那么$β(fg) \leq ω^ξ,$，其中$ξ=\max\{ξ_1+ξ_2, ξ_2+ξ_1}\}.$这些结果不假设所涉及的功能的有界性。

英文摘要：

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. In this paper, 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 $β(f) \leq ω^{ξ_1} \cdot ω^{ξ_2}$ for some countable ordinals $ξ_1$ and $ξ_2$ if and only if there exists a sequence of Baire-1 functions $(f_n)$ converging to $f$ pointwise such that $\sup_nβ(f_n) \leq ω^{ξ_1}$ and $γ((f_n)) \leq ω^{ξ_2}$. We also obtain an extension result for Baire-1 functions analogous to the Tietze Extension Theorem. Finally, it is shown that if $β(f) \leq ω^{ξ_1}$ and $β(g) \leq ω^{ξ_2},$ then $β(fg) \leq ω^ξ,$ where $ξ=\max\{ξ_1+ξ_2, ξ_2+ξ_1}\}.$ These results do not assume the boundedness of the functions involved.

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