Complete nonmeasurability in regular families

TL;DR

The paper demonstrates that in certain regular families of sets within uncountable Polish spaces, there exist subfamilies whose unions are completely nonmeasurable, extending previous results in measure theory.

Contribution

It introduces a generalization showing the existence of completely nonmeasurable unions in regular families of sets within uncountable Polish spaces.

Findings

Existence of subfamilies with completely nonmeasurable unions

Generalization of previous measure-theoretic results

Applicable to families with a Borel base in Polish spaces

Abstract

We show that for a $\sigma $-ideal $\ci$ with a Borel base of subsets of an uncountable Polish space, if $\ca$ is (in several senses) a "regular" family of subsets from $\ci $ then there is a subfamily of $\ca$ whose union is completely nonmeasurable i.e. its intersection with every Borel set not in $\ci $ does not belong to the smallest $\sigma $-algebra containing all Borel sets and $\ci.$ Our results generalize results from \cite{fourpoles} and \cite{fivepoles}.