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

TL;DR

This paper presents a method to extend Baire-one functions defined on certain subsets of perfectly normal spaces to the entire space using a specific series representation, preserving key properties.

Contribution

It introduces a new approximation series for extending Baire-alpha functions from subsets to whole spaces while maintaining their class and norm conditions.

Findings

Extension series converges uniformly on the space.

Extension preserves the Baire-one class and norm conditions.

Method applies to positive functions and Baire-alpha functions.

Abstract

Let $\Omega$ be a perfectly normal topological space, let $A$ be a non-empty $G_\delta$-subset of $\Omega$ 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 $\chi_A$ 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 $F_\sigma$- and $G_\delta$-subsets of $\Omega$ such that the function $\overline{f}\colon\Omega\to\mathbb{R}$ given by the uniformly convergent series on $\Omega$ with the formula: $\overline{f}:=c\sum_{n=0}^\infty\left(\frac{2}{3}\right)^{n+1}\left(\frac{1}{2}-\chi_{H_n}\right)$ extends $f$ with $\overline{f}\in B_1(\Omega)$ and the condition ($\triangle$) of the form: $\|f(A)\|_\infty=\|\overline{f}(\Omega)\|_\infty$. 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 $\Omega$.