First Baire class functions in the pluri-fine topology

Abstract

Let $B_{1}(Ω, \mathbb R)$ be the first Baire class of real functions in the pluri-fine topology on an open set $Ω\subseteq \mathbb C^{n}$ and let $H_{1}^{*}(Ω, \mathbb R)$ be the first functional Lebesgue class of real functions in the same topology. We prove the equality $B_{1}(Ω, \mathbb R)=H_{1}^{*}(Ω, \mathbb R)$ and show that for every $f\in B_{1}(Ω, \mathbb R)$ there is a separately continuous function $g: Ω^{2} \to\mathbb R$ in the pluri-fine topology on $Ω^2$ such that $f$ is the diagonal of $g.$