×

Families of sets related to Rosenthal’s lemma. (English) Zbl 1429.03155

A matrix of non-negative reals \(\langle m^k_n:n,k\in\omega\rangle\) is said to be a Rosenthal matrix, if \(\sum_{n\in\omega}m^k_n\le1\) for every \(k\in\omega\). A family \(\mathcal{F}\subseteq[\omega]^\omega\) is called Rosenthal if for every Rosenthal matrix \(\langle m^k_n:n,k\in\omega\rangle\) and every \(\varepsilon>0\) there exists \(A\in\mathcal{F}\) such that for every \(k\in\omega\), \(\sum_{n\in A\setminus\{k\}}m^k_n<\varepsilon\). This notion was obtained by analysing the proof of Rosenthal’s lemma setting \(m^k_n=\mu_k(a_n)\) for an antichain \(\langle a_n:n\in\omega\rangle\) and a bounded sequence of finitely additive non-negative measures \(\langle\mu_k:k\in\omega\rangle\) in an arbitrary Boolean algebra. In this setting, Rosenthal’s lemma states that \([\omega]^\omega\) is a Rosenthal family. The paper under the review solves the question whether a given family \(\mathcal{F}\) is a Rosenthal family. The author proves the following results: The cardinality of a Rosenthal family cannot be less than the covering of the category \(\text{cov}(\mathcal{M})\) and every base of a selective ultrafilter is a Rosenthal family. Under Martin’s axiom for \(\sigma\)-centered partially ordered sets there exists a non-selective ultrafilter which is a Rosenthal family (in fact it is a P-point that is not a Q-point). The iterated Sacks forcing of length \(\omega_2\) provides a model of ZFC in which there exists a Rosenthal family of cardinality \(<\mathfrak{c}\).

MSC:

03E17 Cardinal characteristics of the continuum
28A33 Spaces of measures, convergence of measures
28A60 Measures on Boolean rings, measure algebras
03E35 Consistency and independence results
03E75 Applications of set theory
05C55 Generalized Ramsey theory
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Argyros, S., Todorčević, S.: Ramsey methods in analysis. In: Advanced Courses in Mathematics, CRM Barcelona. Birkhäuser (2005) · Zbl 1092.46002
[2] Aubrey, J.: Combinatorics for the dominating and unsplitting numbers. J. Symb. Log. 69(2), 482-498 (2004) · Zbl 1069.03038 · doi:10.2178/jsl/1082418539
[3] Bartoszyński, T., Judah, H.: Set Theory: On the Structure of the Real Line. A.K. Peters, Wellesley (1995) · Zbl 0834.04001 · doi:10.1201/9781439863466
[4] Baumgartner, J.E., Laver, R.: Iterated perfect-set forcing. Ann. Math. Log. 17, 271-288 (1979) · Zbl 0427.03043 · doi:10.1016/0003-4843(79)90010-X
[5] Blass, A.: The Rudin-Keisler ordering of P-points. Trans. Am. Math. Soc. 179, 145-166 (1973) · Zbl 0269.02025
[6] Blass, A.: Selective ultrafilters and homogeneity. Ann. Pure Appl. Log. 38(3), 215-255 (1988) · Zbl 0649.03036 · doi:10.1016/0168-0072(88)90027-9
[7] Blass, A.; Foreman, M. (ed.); Kanamori, A. (ed.), Combinatorial cardinal characteristics of the continuum (2010), Dordrecht
[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) · Zbl 1269.03048
[9] Comfort, W.W., Negrepontis, S.: The Theory of Ultrafilters. Die Grundlehren der mathematischen Wissenschaften. Springer, Berlin (1974) · Zbl 0298.02004 · doi:10.1007/978-3-642-65780-1
[10] Diestel, J.: Sequences and Series in Banach Spaces. Springer, Berlin (1984) · Zbl 0542.46007 · doi:10.1007/978-1-4612-5200-9
[11] Diestel, J., Uhl, J.J.: Vector Measures. American Mathematical Society, Providence (1977) · Zbl 0369.46039 · doi:10.1090/surv/015
[12] Graham, R.L., Rothschild, B.L., Spencer, J.H.: Ramsey Theory. Wiley, London (1990) · Zbl 0705.05061
[13] Grigorieff, S.: Combinatorics on ideals and forcing. Ann. Math. Log. 3(4), 363-394 (1971) · Zbl 0328.02041 · doi:10.1016/0003-4843(71)90011-8
[14] Haydon, R.: A nonreflexive Grothendieck space that does not contain \[\ell_\infty\] ℓ∞. Israel J. Math. 40(1), 65-73 (1981) · Zbl 1358.46007 · doi:10.1007/BF02761818
[15] Jech, T.: Set Theory. 3rd Millenium Edition. Springer, Berlin (2002)
[16] Just, W., Weese, M.: Discovering Modern Set Theory, II: Set-Theoretic Tools for Every Mathematician. American Mathematical Society, Providence (1997) · Zbl 0887.03036
[17] Keremedis, K.: On the covering and the additivity number of the real line. Proc. Am. Math. Soc. 123, 1583-1590 (1995) · Zbl 0823.03026 · doi:10.1090/S0002-9939-1995-1234629-6
[18] Komjáth, P., Totik, V.: Problems and Theorems in Classical Set Theory. Springer, Berlin (2006) · Zbl 1103.03041
[19] Koszmider, P., Shelah, S.: Independent families in Boolean algebras with some separation properties. Algebra Universalis 69(4), 305-312 (2013) · Zbl 1283.06018 · doi:10.1007/s00012-013-0227-2
[20] Kunen, K.: Some points in \[\beta{\mathbb{N}}\] βN. Math. Proc. Camb. Philos. Soc. 80(3), 385-398 (1976) · Zbl 0345.02047 · doi:10.1017/S0305004100053032
[21] Kupka, I.: A short proof and generalization of a measure theoretic disjointization lemma. Proc. Am. Math. Soc. 45(1), 70-72 (1974) · Zbl 0291.28004 · doi:10.1090/S0002-9939-1974-0342666-5
[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) · Zbl 0854.04004
[23] Laflamme, C., Leary, C.C.: Filter games on \[\omega\] ω and the dual ideal. Fund. Math. 173(2), 159-173 (2002) · Zbl 0998.03038 · doi:10.4064/fm173-2-4
[24] Miller, A.W.: There are no \[Q\] Q-points in Laver’s model for the Borel conjecture. Proc. Am. Math. Soc. 78(1), 103-106 (1980) · Zbl 0439.03035
[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)
[26] Douwen, EK; Kunen, K. (ed.); Vaughan, JE (ed.), The integers and topology, 111-168 (1984), Amsterdam · doi:10.1016/B978-0-444-86580-9.50006-9
[27] Wimmers, E.L.: The Shelah P-point independence theorem. Israel J. Math. 43(1), 28-48 (1982) · Zbl 0511.03022 · doi:10.1007/BF02761683
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.