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Weak Distributivity, A Problem of Von Neumann and the Mystery of Measurability

Published online by Cambridge University Press:  15 January 2014

Bohuslav Balcar  and
Thomas Jech
Bohuslav Balcar
Affiliation:
Mathematical Institute, AS CR, Zitna 25, CZ-115 67 Praha 1, Czech Republic, E-mail: [email protected], E-mail: [email protected]
Thomas Jech
Affiliation:
Mathematical Institute, AS CR, Zitna 25, CZ-115 67 Praha 1, Czech Republic, E-mail: [email protected], E-mail: [email protected]
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Extract

This article investigates the weak distributivity of Boolean σ-algebras satisfying the countable chain condition. It addresses primarily the question when such algebras carry a σ-additive measure. We use as a starting point the problem of John von Neumann stated in 1937 in the Scottish Book. He asked if the countable chain condition and weak distributivity are sufficient for the existence of such a measure.

Subsequent research has shown that the problem has two aspects: one set theoretic and one combinatorial. Recent results provide a complete solution of both the set theoretic and the combinatorial problems. We shall survey the history of von Neumann's Problem and outline the solution of the set theoretic problem. The technique that we describe owes much to the early work of Dorothy Maharam to whom we dedicate this article.

§1. Complete Boolean algebras and weak distributivity. A Boolean algebra is a set B with Boolean operations a ˅ b (join), a ˄ b (meet) and −a (complement), partial ordering a ≤ b defined by a ˄ b = a and the smallest and greatest element, 0 and 1. By Stone's Representation Theorem, every Boolean algebra is isomorphic to an algebra of subsets of some nonempty set S, under operations a ∪ b, a ∩ b, S − a, ordered by inclusion, with 0 = ∅ and 1 = S.

Complete Boolean algebras and weak distributivity.A Boolean algebra is a set B with Boolean operations a ˅ b (join), a ˄ b (meet) and -a (complement), partial ordering a ≤ b defined by a ˄ b = a and the smallest and greatest element. 0 and 1. By Stone's Representation Theorem, every Boolean algebra is isomorphic to an algebra of subsets of some nonempty set S, under operations a ∪ b, a ∩ b, S -a, ordered by inclusion, with 0 = ϕ and 1 = S.

Type
Articles
Information
Bulletin of Symbolic Logic , Volume 12 , Issue 2 , June 2006 , pp. 241 - 266
Copyright
Copyright © Association for Symbolic Logic 2007

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References

REFERENCES

[1] Abraham, U. and Todorčević, S., Partition properties of ω1 compatible with CH, Fundamenta Mathematicae, vol. 152 (1997), pp. 165–181.Google Scholar
[2] Balcar, B., Franek, F., and Hruška, J., Exhaustive zero-convergence structures on Boolean algebras, Acta Universitatis Carolinae. Mathematica et Physica, vol. 40 (1999), no. 2, pp. 27–41.Google Scholar
[3] Balcar, B., Głowczyński, W., and Jech, T., The sequential topology on complete Boolean algebras, Fundamenta Mathematicae, vol. 155 (1998), no. 1, pp. 59–78.CrossRefGoogle Scholar
[4] Balcar, B., Jech, T., and Pazák, T., Complete ccc Boolean algebras, the order sequential topology and a problem of von Neumann, Preprint, http://arxiv.org/abs/math.LO/0312473, 12 2003.Google Scholar
[5] Balcar, B., Jech, T., and Pazák, T., Complete ccc Boolean algebras, the order sequential topology and a problem of von Neumann, The Bulletin of the London Mathematical Society, vol. 37 (2005), pp. 885–898.Google Scholar
[6] Banach, S. and Kuratowski, K., Sur une Géneralisation du Problème de la Mesure, Fundamenta Mathematicae, vol. 14 (1929), pp. 127–131.Google Scholar
[7] Baumgartner, J. E., Iterated forcing, Surveys in set theory, London Mathematical Society Lecture Note Series, vol. 87, Cambridge University Press, Cambridge, 1983, pp. 1–59.Google Scholar
[8] Farah, I. and Velickovic, B., Von Neumann's problem and large cardinals, The Bulletin of the London Mathematical Society, to appear.Google Scholar
[9] Fremlin, D. H., Measure algebras, Handbook of Boolean algebras, vol. 3, North-Holland Publishing Company, Amsterdam, 1989, pp. 877–980.Google Scholar
[10] Fremlin, D. H., Measure theory, vol. 1–4, Torres Fremlin, Colchester, 2000–2004.Google Scholar
[11] Fremlin, D. H., Problems, Unpublished notes, 01 2004.Google Scholar
[12] Fremlin, D. H., Maharam algebras, Unpublished notes, 12 2004.Google Scholar
[13] Główczyński, W., Measures on Boolean algebras, Proceedings of the American Mathematical Society, vol. 111 (1991), no. 3, pp. 845–849.Google Scholar
[14] Gray, C., Iterated forcing from the strategic point of view, PhD Thesis, Berkeley, 1982.Google Scholar
[15] Hewitt, E. and Ross, K. A., Abstract harmonic analysis, vol. I, Springer-Verlag, Berlin, 1979.Google Scholar
[16] Horn, A. and Tarski, A., Measures in Boolean algebras, Transactions of the American Mathematical Society, vol. 64 (1948), pp. 467–497.Google Scholar
[17] Jech, T., Non-provability of Souslin's hypothesis, Commentationes Mathematicae Universitatis Carolinae, vol. 8 (1967), pp. 291–305.Google Scholar
[18] Jech, T., More game-theoretic properties of Boolean algebras, Annals of Pure and Applied Logic, vol. 26 (1984), pp. 11–29.Google Scholar
[19] Jech, T., Set theory, the third millennium edition, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2002.Google Scholar
[20] Kakutani, S., Über die Metrisation der topologischen Gruppen, Proceedings of the Imperial Academy, Tokyo, vol. 12 (1936), pp. 82–84.Google Scholar
[21] Kalton, N. J. and Roberts, J. W., Uniformly exhaustive submeasures and nearly additive set functions, Transactions of the American Mathematical Society, vol. 278 (1983), pp. 803–816.Google Scholar
[22] Kantorovič, L. V., Vulikh, B. Z., and Pinsker, A. G., Functional analysis in partially ordered spaces, 1950, in Russian.Google Scholar
[23] Kelley, J. L., Measures on Boolean algebras, Pacific Journal of Mathematics, vol. 9 (1959), pp. 1165–1177.Google Scholar
[24] Koppelberg, S., Handbook of Boolean algebras, vol. 1, North-Holland Publishing Company, Amsterdam, 1989.Google Scholar
[25] Kurepa, D., Ensembles ordonnés et ramifiés, Publicationes Mathematicae, University of Belgrade, vol. 4 (1935), pp. 1–138, Zbl. 014.39401.Google Scholar
[26] Maharam, D., On homogeneous measure algebras, Proceedings of the National Academy of Sciences of the United States of America, vol. 28 (1942), pp. 108–111.Google Scholar
[27] Maharam, D., An algebraic characterization of measure algebras, Annals of Mathematics. Second Series, vol. 48 (1947), pp. 154–167.Google Scholar
[28] Mauldin, D. (editor), The Scottish Book, Birkhäuser Boston, Massachusetts, 1981.Google Scholar
[29] Miller, E. W., A note on Souslin's problem, American Journal of Mathematics, vol. 65 (1943), pp. 673–678.Google Scholar
[30] Monk, J. D. (editor), Handbook of Boolean algebras, North-Holland Publishing Company, Amsterdam, 1989.Google Scholar
[31] Quickert, S., CH and the Sacks property, Fundamenta Mathematicae, vol. 171 (2002), no. 1, pp. 93–100.Google Scholar
[32] Shelah, S., Proper forcing, Lecture Notes in Mathematics, vol. 940, Springer-Verlag, Berlin, 1982.Google Scholar
[33] Solovay, R. M. and Tennenbaum, S., Iterated Cohen extensions and Souslin's problem, Annals of Mathematics. Second Series, vol. 94 (1971), pp. 201–245.Google Scholar
[34] Suslin, M., Problème 3, Fundamenta Mathematicae, vol. 1 (1920), p. 223.Google Scholar
[35] Marczewski, E. Szpilrajn, Remarques sur les fonctions complètement additives d'ensemble et sur les ensembles jouissant de la propriété de Baire, Fundamenta Mathematicae, vol. 22 (1934), pp. 303–311.Google Scholar
[36] Talagrand, M., Maharam's problem, Manuscript in preparation, 01 2006.Google Scholar
[37] Tennenbaum, S., Souslin's problem, Proceedings of the National Academy of Sciences of the United States of America, vol. 59 (1968), pp. 60–63.Google Scholar
[38] Todorcevic, S., A dichotomy for P-ideals of countable sets, Fundamenta Mathematicae, vol. 166 (2000), no. 3, pp. 251–267.Google Scholar
[39] Todorcevic, S., A problem of von Neumann and Maharam about algebras supporting continuous submeasures, Fundamenta Mathematicae, vol. 183 (2004), pp. 169–183.Google Scholar
[40] Velickovic, B., CCC forcing and splitting reals, Israel Journal of Mathematics, vol. 147 (2005).Google Scholar
[41] Vladimirov, D. A., Boolean algebras in analysis, Mathematics and its Applications, vol. 540, Kluwer Academic Publishers, Dordrecht, 2002, Translated from the Russian manuscript, Foreword and appendix by Kutateladze, S. S..Google Scholar
[42] von Neumann, J., Continuous geometry, The Institute for Advanced Study, 1936–1937, Notes by L. Roy Wilcox on lectures given during 1935–36 and 1936–37 at the Institute for Advanced Study.Google Scholar
[43] von Neumann, J., Continuous geometry, Princeton University Press, Princeton, New Jersey, 1960.Google Scholar