First Baire class functions in the pluri-fine topology

TL;DR

This paper establishes the equivalence of two classes of first Baire class functions in the pluri-fine topology on complex domains and shows these functions can be represented as diagonals of separately continuous functions.

Contribution

It proves the equality of the first Baire class and the first functional Lebesgue class in the pluri-fine topology and characterizes these functions as diagonals of separately continuous functions.

Findings

Proves $B_{1}(\Omega, \\mathbb R) = H_{1}^{*}(\Omega, \\mathbb R)$.

Shows each first Baire class function is a diagonal of a separately continuous function.

Establishes a link between Baire class functions and separately continuous functions in the pluri-fine topology.

Abstract

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