Covering properties of ideals

• Published in: Archive for mathematical logic 2013-05, Vol.52 (3-4), p.279-294
• Main Authors: Balcerzak, Marek, Farkas, Barnabás, Gła̧b, Szymon
• Format: Article
• Language: English
• Subjects: Algebra; Archives; Covering; Density; Logic; Mathematical logic; Mathematical Logic and Foundations; Mathematics; Mathematics and Statistics; Proving; Theorems
• ISSN: 0933-5846; 1432-0665
• DOI: 10.1007/s00153-012-0316-5

Elekes proved that any infinite-fold cover of a σ -finite measure space by a sequence of measurable sets has a subsequence with the same property such that the set of indices of this subsequence has density zero. Applying this theorem he gave a new proof for the random-indestructibility of the density zero ideal. He asked about other variants of this theorem concerning I -almost everywhere infinite-fold covers of Polish spaces where I is a σ -ideal on the space and the set of indices of the required subsequence should be in a fixed ideal on ω . We introduce the notion of the -covering property of a pair where is a σ -algebra on a set X and is an ideal. We present some counterexamples, discuss the category case and the Fubini product of the null ideal and the meager ideal . We investigate connections between this property and forcing-indestructibility of ideals. We show that the family of all Borel ideals on ω such that has the -covering property consists exactly of non weak Q -ideals. We also study the existence of smallest elements, with respect to Katětov–Blass order, in the family of those ideals on ω such that or has the -covering property. Furthermore, we prove a general result about the cases when the covering property “strongly” fails.