Families of sets with nonmeasurable unions with respect to ideals defined by trees

Abstract

In this note we consider subfamilies of the ideal s ₀ introduced by Marczewski-Szpilrajn and ideals sp ₀, l ₀ analogously defined using complete Laver trees and Laver trees respectively. We show that under some set-theoretical assumptions (${cov(s_0)=\mathfrak{c}}$ for example) in every uncountable Polish space X every family ${\mathcal{A}\subseteq s_0}$ covering X has a subfamily with s-nonmeasurable union. We show the consistency of ${cov(s_0)=\omega_1 < \mathfrak{c}}$ with the existence of a partition ${\mathcal{A}\in[s_0]^{\omega_1}}$ of the real line with a subfamily ${\mathcal{A}'\subseteq\mathcal{A}}$ for which ${\bigcup\mathcal{A}'}$ is s-nonmeasurable. We also show that it is relatively consistent with ZFC that ${\omega_1 < \mathfrak{c}}$ and there exists a maximal almost disjoint family ${\mathcal{A}}$ in the Baire space such that ${\bigcup\mathcal{A}}$ is sp-nonmeasurable. Under CH we show that there is m.a.d. family ${\mathcal{A}}$ in Baire space which is not l-measurable and the same result holds for the ideal sp ₀. Finally we prove the consistency of ${cov(s_0) < \mathfrak{a}}$.