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

Let \(\Omega\) be a perfectly normal topological space, let \(A\) be a non-empty \(G_\delta\)-subset of \(\Omega\) and let \(\mathscr{B}_1(A)\) 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 \(f\in\mathscr {B}_1(A)\) there is a sequence \((H_n)\) of both \(F_\sigma\)- and \(G_\delta\)-subset of \(\Omega\) such that the function \(\overline{f}\colon\Omega\to\mathbb {R}\) given by the uniformly convergent series on \(\Omega\) with the formula: \( \overline{f}:=c\sum_{n=0}^\infty (\frac{2}{3})^{n+1}(\frac{1}{2}-\chi_{H_n}) \) extends \(f\) with \(\overline{f}\in{\mathscr{B}}_1(\Omega)\), \(c=\sup_{x\in\Omega}\lvert{\overline{f}(x)}\rvert\) and the condition \((\triangle)\) of the form: \(\|f\|_\infty=\|\overline{f}\|_\infty\). 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 \(\Omega\).