Abstract
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 \({{\mathcal{J}}}\) on ω. We introduce the notion of the \({{\mathcal{J}}}\) -covering property of a pair \({({\mathcal{A}}, I)}\) where \({{\mathcal{A}}}\) is a σ-algebra on a set X and \({{I \subseteq \mathcal{P}(X)}}\) is an ideal. We present some counterexamples, discuss the category case and the Fubini product of the null ideal \({\mathcal{N}}\) and the meager ideal \({\mathcal{M}}\) . We investigate connections between this property and forcing-indestructibility of ideals. We show that the family of all Borel ideals \({{\mathcal{J}}}\) on ω such that \({\mathcal{M}}\) has the \({{\mathcal{J}}}\) -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 \({\mathcal{J}}\) on ω such that \({\mathcal{N}}\) or \({\mathcal{M}}\) has the \({\mathcal{J}}\) -covering property. Furthermore, we prove a general result about the cases when the covering property “strongly” fails.
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The first author was supported by the Polish Ministry of Science and Higher Education Grant No. N N201 414939 (2010–2013); the second author was supported by Hungarian National Foundation for Scientific Research grant nos. 68262, 83726 and 77476; the third author was supported by the Polish Ministry of Science and Higher Education Grant No. IP2011 014671 (2012–2014).
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Balcerzak, M., Farkas, B. & Gła̧b, S. Covering properties of ideals. Arch. Math. Logic 52, 279–294 (2013). https://doi.org/10.1007/s00153-012-0316-5
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DOI: https://doi.org/10.1007/s00153-012-0316-5
Keywords
- Ideals
- Infinite-fold covers
- Covering properties
- Forcing-indestructibility
- Meager sets
- Null sets
- Katětov–Blass order
- Borel determinacy
- Borel ideals
- P-ideals
- Fubini product