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Compactness of Lobe spaces

Published online by Cambridge University Press:  12 March 2014

Renling Jin
Department of Mathematics, College of Charleston, Charleston, SC 29424, USA. Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, USA.
Saharon Shelah
Institute of Mathematics, The Hebrew University, Jerusalem, Israel. Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, USA. Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA.


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].

Research Article
Copyright © Association for Symbolic Logic 1998

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[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. 173175.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. 975987.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. 12331242.Google Scholar
[11] Ross, D. A., Compact measures have Loeb preimages, Proceedings of the American Mathematical Society, vol. 115 (1992), pp. 365370.CrossRefGoogle 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. 178185.Google Scholar
[13] Ross, D. A., Unions of Loeb nullsets, Proceedings of the American Mathematical Society, vol. 124 (1996), pp. 18831888.CrossRefGoogle 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. 7789.CrossRefGoogle Scholar
[16] Stroyan, K. D. and Bayod, J. M., Foundation of infinitesimal stochastic analysis, North-Holland, Amsterdam, 1986.Google Scholar