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Maharam algebras and Cohen reals


Authors: Ilijas Farah and Boban Velickovic
Journal: Proc. Amer. Math. Soc. 135 (2007), 2283-2290
MSC (2000): Primary 03Exx; Secondary 28Axx
DOI: https://doi.org/10.1090/S0002-9939-07-08759-X
Published electronically: February 2, 2007
MathSciNet review: 2299506
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Abstract: We show that the product of any two nonatomic Maharam algebras adds a Cohen real. As a corollary of this and a result of Shelah (1994) we obtain that the product of any two nonatomic ccc Souslin forcing notions adds a Cohen real.


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  • 1. U. Abraham and S. Todorcevic, Partition properties of $ \omega\sb 1$ compatible with CH, Fund. Math. 152 (1997), no. 2, 165-181. MR 1441232 (98b:03064)
  • 2. B. Balcar, T. Jech, and T. Pazák, Complete ccc Boolean algebras, the order sequential topology, and a problem of von Neumann, Bulletin of the London Math. Soc. 37 (2005), no. 6, 885-898. MR 2186722
  • 3. T. Bartoszynski and H. Judah, Set theory: on the structure of the real line, A.K. Peters, 1995. MR 1350295 (96k:03002)
  • 4. J.P.R. Christensen, Some results with relation to the control measure problem, Vector space measures and applications II (R.M. Aron and S. Dineen, eds.), Lecture Notes in Mathematics, vol. 645, Springer, 1978, pp. 27-34. MR 0502425 (80e:28008)
  • 5. I. Farah and J. Zapletal, Between Maharam's and von Neumann's problems, Math. Res. Lett. 11 (2004), no. 5-6, 673-684. MR 2106234 (2005i:03059)
  • 6. D.H. Fremlin, Measure theory, vol. 3, Torres-Fremlin, 2002.
  • 7. -, Maharam algebras, preprint, University of Essex, available at http://www.essex.ac.uk/maths/staff/fremlin/preprints.htm, 2004.
  • 8. Thomas Jech, Set theory, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003, The third millennium edition, revised and expanded. MR 1940513 (2004g:03071)
  • 9. Nigel Kalton, The Maharam problem, Séminaire d'initiation à l'analyse, Publ. Math. Univ. Pierre et Marie Curie, vol. 94, Univ. Paris VI, Paris, 1989, Exp. No. 18, 13 pp. MR 1107322 (92c:28005)
  • 10. N.J. Kalton and J.W. Roberts, Uniformly exhaustive submeasures and nearly additive set functions, Transactions of the American Mathematical Society 278 (1983), 803-816. MR 0701524 (85f:28006)
  • 11. A. Louveau, Progrès récents sur le problème de Maharam d'après N.J. Kalton et J.W. Roberts, Séminaire d'initiation à l'analyse, 28ème année (G. Choquet et al., ed.), no. 20, 1983/84, pp. 01-08.
  • 12. Dorothy Maharam, An algebraic characterization of measure algebras, Ann. of Math. (2) 48 (1947), 154-167. MR 0018718 (8:321b)
  • 13. J.W. Roberts, Maharam's problem, Proceedings of the Orlitz memorial conference (P. Kranz and I. Labuda, eds.), 1991, unpublished.
  • 14. S. Shelah, How special are Cohen and random forcings, i.e. Boolean algebras of the family of subsets of reals modulo meagre or null, Israel Journal of Mathematics 88 (1994), 159-174. MR 1303493 (96g:03090)
  • 15. Otmar Spinas, Ramsey and freeness properties of Polish planes, Proc. London Math. Soc. (3) 82 (2001), no. 1, 31-63. MR 1794256 (2001j:03088)
  • 16. Michel Talagrand, Maharam's problem, preprint, 31 pages, 2006.
  • 17. S. Todorcevic, A dichotomy for P-ideals of countable sets, Fundamenta Mathematicae 166 (2000), 251-267. MR 1809418 (2001k:03111)
  • 18. B. Velickovic, The basis problem for ccc posets, Set Theory, DIMACS Series, vol. 58, American Mathematical Society, 2002, pp. 149-160. MR 1903857 (2003e:03098)
  • 19. -, CCC forcings and splitting reals, Israel Journal of Mathematics 147 (2005), 209-220. MR 2166361 (2006d:03088)
  • 20. Boban Velickovic and W. Hugh Woodin, Complexity of reals in inner models of set theory, Ann. Pure Appl. Logic 92 (1998), no. 3, 283-295. MR 1640916 (99f:03067)

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Additional Information

Ilijas Farah
Affiliation: Department of Mathematics and Statistics, York University, 4700 Keele Street, North York, Ontario, Canada M3J 1P3
Address at time of publication: Matematički Institut, Kneza Mihaila 35, Beograd, Serbia and Montenegro
Email: ifarah@mathstat.yorku.ca

Boban Velickovic
Affiliation: Equipe de Logique Mathématique, UFR de Mathématiques (case 7012), Université Denis-Diderot Paris 7, 2 place Jussieu, 75251 Paris Cedex 05, France
Address at time of publication: Matematički Institut, Kneza Mihaila 35, Beograd, Serbia and Montenegro
Email: boban@logique.jussieu.fr

DOI: https://doi.org/10.1090/S0002-9939-07-08759-X
Keywords: Exhaustive submeasure, Maharam algebra, Cohen real
Received by editor(s): March 8, 2006
Published electronically: February 2, 2007
Additional Notes: These results were obtained in December 2004 while the authors were visiting the E. Schrödinger Institute in Vienna. We would like to thank the Institute for providing hospitality and a stimulating environment. The first author was partially supported by NSERC
Communicated by: Julia Knight
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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