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 \documentclass[12pt]{minimal}
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\begin{document}$${{\mathcal{J}}}$$\end{document} on ω. We introduce the notion of the \documentclass[12pt]{minimal}
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\begin{document}$${{\mathcal{J}}}$$\end{document} -covering property of a pair \documentclass[12pt]{minimal}
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\begin{document}$${({\mathcal{A}}, I)}$$\end{document} where \documentclass[12pt]{minimal}
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\begin{document}$${{\mathcal{A}}}$$\end{document} is a σ-algebra on a set X and \documentclass[12pt]{minimal}
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\begin{document}$${{I \subseteq \mathcal{P}(X)}}$$\end{document} is an ideal. We present some counterexamples, discuss the category case and the Fubini product of the null ideal \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{N}}$$\end{document} and the meager ideal \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{M}}$$\end{document} . We investigate connections between this property and forcing-indestructibility of ideals. We show that the family of all Borel ideals \documentclass[12pt]{minimal}
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\begin{document}$${{\mathcal{J}}}$$\end{document} on ω such that \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{M}}$$\end{document} has the \documentclass[12pt]{minimal}
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\begin{document}$${{\mathcal{J}}}$$\end{document} -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 \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{J}}$$\end{document} on ω such that \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{N}}$$\end{document} or \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{M}}$$\end{document} has the \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{J}}$$\end{document} -covering property. Furthermore, we prove a general result about the cases when the covering property “strongly” fails.
