Maharam algebras and Cohen reals

Farah, Ilijas; Veličković, Boban

doi:10.1090/S0002-9939-07-08759-X

Maharam algebras and Cohen reals
HTML articles powered by AMS MathViewer

by Ilijas Farah and Boban Veličković PDF

Proc. Amer. Math. Soc. 135 (2007), 2283-2290 Request permission

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.

References

Uri Abraham and Stevo Todorčević, Partition properties of $\omega _1$ compatible with CH, Fund. Math. 152 (1997), no. 2, 165–181. MR 1441232, DOI 10.4064/fm-152-2-165-181
B. Balcar, T. Jech, and T. Pazák, Complete CCC Boolean algebras, the order sequential topology, and a problem of von Neumann, Bull. London Math. Soc. 37 (2005), no. 6, 885–898. MR 2186722, DOI 10.1112/S0024609305004807
Tomek Bartoszyński and Haim Judah, Set theory, A K Peters, Ltd., Wellesley, MA, 1995. On the structure of the real line. MR 1350295, DOI 10.1201/9781439863466
Jens Peter Reus Christensen, Some results with relation to the control measure problem, Vector space measures and applications (Proc. Conf., Univ. Dublin, Dublin, 1977) Lecture Notes in Math., vol. 645, Springer, Berlin-New York, 1978, pp. 27–34. MR 502425
Ilijas Farah and Jindřich Zapletal, Between Maharam’s and von Neumann’s problems, Math. Res. Lett. 11 (2004), no. 5-6, 673–684. MR 2106234, DOI 10.4310/MRL.2004.v11.n5.a10
D.H. Fremlin, Measure theory, vol. 3, Torres–Fremlin, 2002.
—, Maharam algebras, preprint, University of Essex, available at http://www.essex.ac.uk/maths/staff/fremlin/preprints.htm, 2004.
Thomas Jech, Set theory, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. The third millennium edition, revised and expanded. MR 1940513
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, pp. Exp. No. 18, 13. MR 1107322
N. J. Kalton and James W. Roberts, Uniformly exhaustive submeasures and nearly additive set functions, Trans. Amer. Math. Soc. 278 (1983), no. 2, 803–816. MR 701524, DOI 10.1090/S0002-9947-1983-0701524-4
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.
Dorothy Maharam, An algebraic characterization of measure algebras, Ann. of Math. (2) 48 (1947), 154–167. MR 18718, DOI 10.2307/1969222
J.W. Roberts, Maharam’s problem, Proceedings of the Orlitz memorial conference (P. Kranz and I. Labuda, eds.), 1991, unpublished.
Saharon Shelah, How special are Cohen and random forcings, i.e. Boolean algebras of the family of subsets of reals modulo meagre or null, Israel J. Math. 88 (1994), no. 1-3, 159–174. MR 1303493, DOI 10.1007/BF02937509
Otmar Spinas, Ramsey and freeness properties of Polish planes, Proc. London Math. Soc. (3) 82 (2001), no. 1, 31–63. MR 1794256, DOI 10.1112/S0024611500012740
Michel Talagrand, Maharam’s problem, preprint, 31 pages, 2006.
Stevo Todorčević, A dichotomy for P-ideals of countable sets, Fund. Math. 166 (2000), no. 3, 251–267. MR 1809418, DOI 10.4064/fm-166-3-251-267
Boban Velickovic, The basis problem for CCC posets, Set theory (Piscataway, NJ, 1999) DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 58, Amer. Math. Soc., Providence, RI, 2002, pp. 149–160. MR 1903857, DOI 10.1090/dimacs/058/12
Boban Velickovic, ccc forcing and splitting reals, Israel J. Math. 147 (2005), 209–220. MR 2166361, DOI 10.1007/BF02785365
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, DOI 10.1016/S0168-0072(98)00010-4