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 $\Omega$.
Categories: math.CA