Remarks on nonmeasurable unions of big point families

TL;DR

This paper investigates conditions under which unions of large point families in uncountable Polish spaces are nonmeasurable, with applications to null sets and first category sets in real analysis.

Contribution

It establishes new conditions ensuring the existence of nonmeasurable unions within certain large point families, extending classical results.

Findings

Existence of nonmeasurable unions under specific conditions

Application to null and first category ideals in real line

Generalization of classical nonmeasurability results

Abstract

We show that under some conditions on a family $\mathcal{A}\subset\bbi$ there exists a subfamily $\mathcal{A}_0\subset\mathcal{A}$ such that $\bigcup \mathcal{A}_0$ is nonmeasurable with respect to a fixed ideal $\bbi$ with Borel base of a fixed uncountable Polish space. Our result applies to the classical ideal of null subsets of the real line and to the ideal of first category subsets of the real line.