An Extension of the Baire Property

TL;DR

This paper introduces the extended Baire property sets for Polish spaces, explores their properties, and demonstrates their usefulness in analysis, including a generalized Pettis Theorem leading to new automatic continuity results.

Contribution

It defines the $EBP(X)$-sets for Polish spaces, studies their properties, and proves a generalized Pettis Theorem with applications in automatic continuity.

Findings

$EBP(X)$ contains $BP(X)$ and may be strictly larger.

A generalized Pettis Theorem is established.

New automatic continuity results for Polish groups are derived.

Abstract

The purpose of this paper is to define for every Polish space $X$ a class of sets, the $EBP(X)$-sets or the extended Baire property sets, to work out many properties of the $EBP(X)$-sets and to show their usefulness in analysis. For example, a proper generalization of the Pettis Theorem is proved in this context that furnishes a new automatic continuity result for Polish groups. The name extended Baire property sets is reasonable since $EBP(X)$ contains the Baire property sets $BP(X)$ and it is consistent with ZFC that the containment is proper.