Abstract
A family \(\mathcal {F}\subseteq \left[ \omega \right] ^\omega \) is called Rosenthal if for every Boolean algebra \(\mathcal {A}\), bounded sequence \(\big \langle \mu _k:\ k\in \omega \big \rangle \) of measures on \(\mathcal {A}\), antichain \(\big \langle a_n:\ n\in \omega \big \rangle \) in \(\mathcal {A}\), and \(\varepsilon >0\), there exists \(A\in \mathcal {F}\) such that \(\sum _{n\in A, n\ne k}\mu _k(a_n)<\varepsilon \) for every \(k\in A\). Well-known and important Rosenthal’s lemma states that \(\left[ \omega \right] ^\omega \) is a Rosenthal family. In this paper we provide a necessary condition in terms of antichains in \({\wp (\omega )}\) for a family to be Rosenthal which leads us to a conclusion that no Rosenthal family has cardinality strictly less than \({{\mathrm{cov}}}(\mathcal {M})\), the covering of category. We also study ultrafilters on \(\omega \) which are Rosenthal families—we show that the class of Rosenthal ultrafilters contains all selective ultrafilters (and consistently selective ultrafilters comprise a proper subclass).
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Open access funding provided by Austrian Science Fund (FWF). The results of the paper come partially from author’s Ph.D. thesis [25] written under the supervision of Piotr Koszmider, whom the author would like to thank for the guidance, inspiring discussions and helpful comments.
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The author was supported by the FWF Grant I 2374-N35.
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Sobota, D. Families of sets related to Rosenthal’s lemma. Arch. Math. Logic 58, 53–69 (2019). https://doi.org/10.1007/s00153-018-0621-8
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DOI: https://doi.org/10.1007/s00153-018-0621-8