Archive for Mathematical Logic

, Volume 58, Issue 1–2, pp 53–69 | Cite as

Families of sets related to Rosenthal’s lemma

  • Damian SobotaEmail author
Open Access


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).


Rosenthal’s lemma Ultrafilters Selective ultrafilters P-points Q-points 

Mathematics Subject Classification

Primary 28A33 28A60 03E17 Secondary 03E35 03E75 05C55 



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.


  1. 1.
    Argyros, S., Todorčević, S.: Ramsey methods in analysis. In: Advanced Courses in Mathematics, CRM Barcelona. Birkhäuser (2005)Google Scholar
  2. 2.
    Aubrey, J.: Combinatorics for the dominating and unsplitting numbers. J. Symb. Log. 69(2), 482–498 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bartoszyński, T., Judah, H.: Set Theory: On the Structure of the Real Line. A.K. Peters, Wellesley (1995)CrossRefzbMATHGoogle Scholar
  4. 4.
    Baumgartner, J.E., Laver, R.: Iterated perfect-set forcing. Ann. Math. Log. 17, 271–288 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Blass, A.: The Rudin–Keisler ordering of P-points. Trans. Am. Math. Soc. 179, 145–166 (1973)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Blass, A.: Selective ultrafilters and homogeneity. Ann. Pure Appl. Log. 38(3), 215–255 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Blass, A.: Combinatorial cardinal characteristics of the continuum. In: Foreman, M., Kanamori, A. (eds.) Handbook of Set Theory. Springer, Dordrecht (2010)Google Scholar
  8. 8.
    Blass, A.: Ultrafilters and set theory. In: Bergelson, V., Blass, A., Nasso, M., Jin, R. (eds.) Ultrafilters Across Mathematics. Contemporary Mathematics, vol. 530, pp 49–71. American Mathematical Society, Providence (2010)Google Scholar
  9. 9.
    Comfort, W.W., Negrepontis, S.: The Theory of Ultrafilters. Die Grundlehren der mathematischen Wissenschaften. Springer, Berlin (1974)CrossRefzbMATHGoogle Scholar
  10. 10.
    Diestel, J.: Sequences and Series in Banach Spaces. Springer, Berlin (1984)CrossRefzbMATHGoogle Scholar
  11. 11.
    Diestel, J., Uhl, J.J.: Vector Measures. American Mathematical Society, Providence (1977)CrossRefzbMATHGoogle Scholar
  12. 12.
    Graham, R.L., Rothschild, B.L., Spencer, J.H.: Ramsey Theory. Wiley, London (1990)zbMATHGoogle Scholar
  13. 13.
    Grigorieff, S.: Combinatorics on ideals and forcing. Ann. Math. Log. 3(4), 363–394 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Haydon, R.: A nonreflexive Grothendieck space that does not contain \(\ell _\infty \). Israel J. Math. 40(1), 65–73 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Jech, T.: Set Theory. 3rd Millenium Edition. Springer, Berlin (2002)Google Scholar
  16. 16.
    Just, W., Weese, M.: Discovering Modern Set Theory, II: Set-Theoretic Tools for Every Mathematician. American Mathematical Society, Providence (1997)zbMATHGoogle Scholar
  17. 17.
    Keremedis, K.: On the covering and the additivity number of the real line. Proc. Am. Math. Soc. 123, 1583–1590 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Komjáth, P., Totik, V.: Problems and Theorems in Classical Set Theory. Springer, Berlin (2006)zbMATHGoogle Scholar
  19. 19.
    Koszmider, P., Shelah, S.: Independent families in Boolean algebras with some separation properties. Algebra Universalis 69(4), 305–312 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Kunen, K.: Some points in \(\beta {\mathbb{N}}\). Math. Proc. Camb. Philos. Soc. 80(3), 385–398 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Kupka, I.: A short proof and generalization of a measure theoretic disjointization lemma. Proc. Am. Math. Soc. 45(1), 70–72 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Laflamme, C.: Filter games and combinatorial properties of strategies. In: Bartoszyński, T., Scheepers, M. (eds.) Set Theory. Contemporary Mathematics, vol. 192, pp. 51–67. American Mathematical Society, Providence (1996)Google Scholar
  23. 23.
    Laflamme, C., Leary, C.C.: Filter games on \(\omega \) and the dual ideal. Fund. Math. 173(2), 159–173 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Miller, A.W.: There are no \(Q\)-points in Laver’s model for the Borel conjecture. Proc. Am. Math. Soc. 78(1), 103–106 (1980)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Sobota, D.: Cardinal invariants of the continuum and convergence of measures on compact spaces. Ph.D. thesis, Institute of Mathematics, Polish Academy of Sciences (2016)Google Scholar
  26. 26.
    van Douwen, E.K.: The integers and topology. In: Kunen, K., Vaughan, J.E. (eds.) Handbook of Set-Theoretic Topology, pp. 111–168. North-Holland, Amsterdam (1984)CrossRefGoogle Scholar
  27. 27.
    Wimmers, E.L.: The Shelah P-point independence theorem. Israel J. Math. 43(1), 28–48 (1982)MathSciNetCrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Kurt Gödel Research Center for Mathematical LogicUniversität WienWienAustria

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