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

Let $\mathrm{\Omega }$ be a perfectly normal topological space, let $A$ be a non-empty $G_{\delta }$-subset of $\mathrm{\Omega }$ and let $B_1(A)$ denote the space of all functions $A\to \mathbb{R}$ of Baire-one class on $A$. Let also $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{\mathrm{\infty }}$ 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 $\mathrm{\Omega }$ such that the function $\bar{f}\text{\hspace{0.17em}⁣}:\mathrm{\Omega }\to \mathbb{R}$ given by the uniformly convergent series on $\mathrm{\Omega }$ with the formula: $\bar{f}:=c{\sum }_{n=0}^{\mathrm{\infty }}{\left(\frac{2}{3}\right)}^{n+1}(\frac{1}{2}-{\chi }_{H_n})$ extends $f$ with $\bar{f}\in B_1(\mathrm{\Omega })$ and the condition ($\mathrm{△}$) of the form: $\mathrm{\parallel }f(A){\mathrm{\parallel }}_{\mathrm{\infty }}=\mathrm{\parallel }\bar{f}(\mathrm{\Omega }){\mathrm{\parallel }}_{\mathrm{\infty }}$. We apply the above series to obtain an extension of $f$ positive to $\bar{f}$ positive with the condition ($\mathrm{△}$). A similar technique allows us to obtain an extension of Baire-alpha function on $A$ to Baire-alpha function on $\mathrm{\Omega }$.