Note on $s_0$ nonmeasurable unions

TL;DR

This paper investigates the measurability properties of unions of families of sets within the $s_0$ ideal in Polish spaces, demonstrating various consistency results related to set-theoretic assumptions and the existence of nonmeasurable unions.

Contribution

It establishes the consistency of certain set-theoretic conditions under which unions of $s_0$-ideal sets can be nonmeasurable, extending understanding of measure-theoretic properties in descriptive set theory.

Findings

Existence of nonmeasurable unions under $cov(s_0)=\mathfrak{c}$

Construction of partitions with $s$-nonmeasurable unions in the real line

Consistency results involving $\omega_1$, $\mathfrak{c}$, and mad families

Abstract

In this note we consider an arbitrary families of sets of $s_0$ ideal introduced by Marczewski-Szpilrajn. We show that in any uncountable Polish space $X$ and under some combinatorial and set theoretical assumptions ($cov(s_0)=\c$ for example), that for any family $\ca\subseteq s_0$ with $\bigcup\ca =X$, we can find a some subfamily $\ca'\subseteq\ca$ such that the union $\bigcup\ca'$ is not $s$-measurable. We have shown a consistency of the $cov(s_0)=\omega_1<\c$ and existence a partition of the size $\omega_1$ $\ca\in [s_0]^{\omega}$ of the real line $\bbr$, such that there exists a subfamily $\ca'\subseteq\ca$ for which $\bigcup\ca'$ is $s$-nonmeasurable. We also showed that it is relatively consistent with ZFC theory that $\omega_1<\c$ and existence of m.a.d. family $\ca$ such that $\bigcup\ca$ is $s$-nonmeasurable in Cantor space $2^\omega$ or Baire space $\omega^\omega$. The consistency of $a<cov(s_0)$ and $cov(s_0)<a$ is proved also.