Completely nonmeasurable unions

TL;DR

Under certain set-theoretic assumptions, the paper proves that for specific ideals on Polish spaces, there exist uncountable disjoint subfamilies whose unions are completely nonmeasurable, extending the Four Poles Theorem.

Contribution

The paper generalizes the Four Poles Theorem by establishing the existence of completely nonmeasurable unions under broader conditions involving c.c.c. ideals and set-theoretic assumptions.

Findings

Existence of uncountable disjoint subfamilies with nonmeasurable unions

Generalization of the Four Poles Theorem to broader ideals

Results depend on the absence of smaller quasi-measurable cardinals

Abstract

Assume that there is no quasi-measurable cardinal smaller than $2^\omega$. ($\kappa$ is quasi measurable if there exists $\kappa $-additive ideal $\ci $ of subsets of $\kappa $ such that the Boolean algebra $P(\kappa)/\ci$ satisfies c.c.c.) We show that for a c.c.c. $\sigma $-ideal I with a Borel base of subsets of an uncountable Polish space, if $\cal A$ is a point-finite family of subsets from I then there is an uncountable collection of pairwise disjoint subfamilies of $\cal A$ whose union is completely nonmeasurable i.e. its intersection with every non-small Borel set does not belong to the $\sigma $-field generated by Borel sets and the ideal I. This result is a generalization of Four Poles Theorem.