An approximation form of the Kuratowski Extension Theorem for Baire-alpha functions final version

Abstract

Let $Ω$ be a perfectly normal topological space, let $A$ be a non-empty $G_δ$-subset of $Ω$ and let $B_1(A)$ denote the space of all functions $A\to\mathbb{R}$ of Baire-one class on $A$. Let also $\|\cdot\|_\infty$ be the supremum norm. The symbol $χ_A$ stands for the characteristic function of $A$. We prove that for every bounded function $f\in B_1(A)$ there is a sequence $(H_n)$ of both $F_σ$- and $G_δ$-subsets of $Ω$ such that the function $\overline{f}\colonΩ\to\mathbb{R}$ given by the uniformly convergent series on $Ω$ with the formula: $\overline{f}:=c\sum_{n=0}^\infty\left(\frac{2}{3}\right)^{n+1}\left(\frac{1}{2}-χ_{H_n}\right)$ extends $f$ with $\overline{f}\in B_1(Ω)$ and the condition ($\triangle$) of the form: $\|f(A)\|_\infty=\|\overline{f}(Ω)\|_\infty$. We apply the above series to obtain an extension of $f$ positive to $\overline{f}$ positive with the condition ($\triangle$). A similar technique allows us to obtain an extension of Baire-alpha function on $A$ to Baire-alpha function on $Ω$.