Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-25T18:11:33.487Z Has data issue: false hasContentIssue false
  • English
  • Français

Compactness of Lobe spaces

Published online by Cambridge University Press:  12 March 2014

Renling Jin  and
Saharon Shelah
Renling Jin
Affiliation:
Department of Mathematics, College of Charleston, Charleston, SC 29424, USA. E-mail:jinr@cofc.edu Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, USA. E-mail:jinr@cofc.edu
Saharon Shelah
Affiliation:
Institute of Mathematics, The Hebrew University, Jerusalem, Israel. E-mail:shelah@math.huji.ac.il Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, USA. E-mail:shelah@math.huji.ac.il Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA. E-mail:shelah@math.huji.ac.il
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we show that the compactness of a Loeb space depends on its cardinality, the nonstandard universe it belongs to and the underlying model of set theory we live in. In §1 we prove that Loeb spaces are compact under various assumptions, and in §2 we prove that Loeb spaces are not compact under various other assumptions. The results in §1 and §2 give a quite complete answer to a question of D.Ross in [9], [11] and [12].

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 63 , Issue 4 , December 1998 , pp. 1371 - 1392
Copyright
Copyright © Association for Symbolic Logic 1998

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Albeverio, S., Fenstad, J. E., Hoegh-Krohn, R., and Lindstrom, T., Nonstandard methods in stochastic analysis and mathematical physics, Academic Press, Orlando, 1986.Google Scholar
[2] Aldaz, J., On compactness and Loeb measures, Proceedings of the American Mathematical Society, vol. 123 (1995), pp. 173—175.CrossRefGoogle Scholar
[3] Chang, C. C. and Keisler, H. J., Model theory, North-Holland, 1973, third edition, 1990.Google Scholar
[4] Fremlin, D., Measure algebra, Handbook of boolean algebra (Monk, J. D. and Bonnet, R., editors), vol. 3, North-Holland, Amsterdam, 1989.Google Scholar
[5] Jin, R., The isomorphism property versus the special model axiom, this Journal, vol. 57 (1992), no. 3, pp. 975—987.Google Scholar
[6] Kunen, K., Set theory—an introduction to independence proofs, North-Holland, Amsterdam, 1980.Google Scholar
[7] Kunen, K., Random and Cohen reals, Handbook of set theoretic topology (Kunen, K. and Vaughan, J. E., editors), North-Holland, Amsterdam, 1984.Google Scholar
[8] Lindstrom, T., An imitation to nonstandard analysis, Nonstandard analysis and its application (Cutland, N., editor), Cambridge University Press, 1988.Google Scholar
[9] Ross, D. A., Measurable transformations in saturated models of analysis, Ph.D. thesis , University of Wisconsin-Madison, 1983.Google Scholar
[10] Ross, D. A., The special model axiom in nonstandard analysis, this Journal, vol. 55 (1990), pp. 1233—1242.Google Scholar
[11] Ross, D. A., Compact measures have Loeb preimages, Proceedings of the American Mathematical Society, vol. 115 (1992), pp. 365—370.Google Scholar
[12] Ross, D. A., Unions of Loeb nullsets: the context, Developments in nonstandard mathematics (Cutland, N. J. et al., editors), Pitman Research Notes in Mathematics, vol. 336, 1995, pp. 178—185.Google Scholar
[13] Ross, D. A., Unions of Loeb nullsets, Proceedings of the American Mathematical Society, vol. 124 (1996), pp. 1883—1888.Google Scholar
[14] Shelah, S., Cellularity offree products of Boolean algebra, Fundamenta Mathematicae, to appear.Google Scholar
[15] Shelah, S., Remarks on Boolean algebra, Algebra Universalis, vol. 11 (1980), pp. 77—89.Google Scholar
[16] Stroyan, K. D. and Bayod, J. M., Foundation of infinitesimal stochastic analysis, North-Holland, Amsterdam, 1986.Google Scholar