The Set-Cover game and nonmeasurable unions

TL;DR

This paper generalizes a classical result on non-measurable unions using game theory and explores applications to ideals, measurable functions, and topological group homomorphisms.

Contribution

It introduces a game-theoretic framework to extend classical non-measurable union results and applies it to various areas in measure theory and topology.

Findings

Generalized non-measurable union results using game theory

Applied results to Marczewski--Burstin ideals

Established countability and continuity properties of measurable functions

Abstract

Using a game-theoretic approach we present a generalization of the classical result of Brzuchowski, Cicho\'n, Grzegorek and Ryll-Nardzewski on non-measurable unions. We also present applications of obtained results to Marczewski--Burstin representable ideals, as well as to establishing some countability and continuity properties of measurable functions and homomorphisms between topological groups.