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

Content Provider	Springer Nature Link
Author	Rałowski, Robert
Copyright Year	2015
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 $${cov(s_0) < \mathfrak{a}}$$ .
Ending Page	658
Page Count	10
Starting Page	649
File Format	PDF
ISSN	09335846
e-ISSN	14320665
Journal	Archive for Mathematical Logic
Issue Number	5-6
Volume Number	54
Language	English
Publisher	Springer Berlin Heidelberg
Publisher Date	2015-05-20
Publisher Place	Berlin, Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Complete Laver tree Descriptive set theory Algebra Classes of sets (Borel fields, $\sigma$ -rings, etc.), measurable sets, Suslin sets, analytic sets Mathematics Marczewski ideal m.a.d. family Nonmeasurable set Mathematical Logic and Foundations Consistency and independence results
Content Type	Text
Resource Type	Article
Subject	Philosophy Logic

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