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Ultrafilters which extend measures

Published online by Cambridge University Press:  12 March 2014

Michael Benedikt
Michael Benedikt*
Affiliation:
Bell Laboratories, 1000 E. Warrenville RD., Naperville, IL 60566, USA, E-mail: benedikt@research.bell-labs.com
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Abstract

We study classes of ultrafilters on ω defined by a natural property of the Loeb measure in the Nonstandard Universe corresponding to the ultrafilter. This class, the Property M ultrafilters, is shown to contain all ultrafilters built up by taking iterated products over collections of pairwise nonisomorphic selective ultrafilters. Results on Property M ultrafilters are applied to the construction of extensions of probability measures, and to the study of measurable reductions between ultrafilters.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 63 , Issue 2 , June 1998 , pp. 638 - 662
Copyright
Copyright © Association for Symbolic Logic 1998

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References

REFERENCES

[1] Albeverio, S., Fenstad, J., Høegh-Krohn, R., and Lindstrøm, T., Nonstandard methods in stochastic analysis and mathematical physics, Academic Press, 1986.Google Scholar
[2] Benedikt, M., Nonstandard analysis and special ultrafilters, Ph.D. thesis , University of Wisconsin, 1993.Google Scholar
[3] Blass, A., A model-theoretic view of some special ultrafilters, Logic colloquium '77 (Macintyre, , Pacholski, , and Paris, , editors), North-Holland, Amsterdam, 1978.Google Scholar
[4] Blass, A., Kleene degrees of ultrafilters, The Kleene symposium (Barwise, , Keisler, , and Kunen, , editors), North-Holland, Amsterdam, 1980.Google Scholar
[5] Blass, A., Selective ultrafilters and homogeneity, Annals of Pure and Applied Logic, vol. ? (1989), pp. 252–285.Google Scholar
[6] Booth, D., Ultrafilters on a countable set, Annals of Mathematical Logic, vol. 2 (1970), pp. 1–24.10.1016/0003-4843(70)90005-7Google Scholar
[7] Chang, C.C. and Keisler, H.J., Model theory, 3rd ed., North-Holland, Amsterdam, 1991.Google Scholar
[8] Daguenet, M., Ultrafilters a la Façon Ramsey, Transactions of the American Mathematical Society, vol. 250 (1979), pp. 91–120.Google Scholar
[9] Fremlin, D. H., unpublished manuscript, 1987.Google Scholar
[10] Fremlin, D. H., Measure algebras, Handbook of Boolean algebras (Monk, , editor), North-Holland, Amsterdam, 1989.Google Scholar
[11] Henson, C. W. and Wattenberg, B., Egoroff's theorem and the distribution of standard points in a nonstandard model, Proceedings of the American Mathematical Society, vol. 81 (1981), pp. 455–461.Google Scholar
[12] Hoover, D. N. and Keisler, H. J., Adapted probability distributions, Transactions of the American Mathematical Society, vol. 286 (1984), pp. 159–201.Google Scholar
[13] Keisler, H. J., An infinitesimal approach to stochastic analysis, Memoirs of the American Mathematical Society, no. 297, 1984.10.1090/memo/0297Google Scholar
[14] Loeb, P. A., Conversion from nonstandard to standard measure spaces and applications in probability theory, Transactions of the American Mathematical Society, vol. 81 (1975),pp. 113–122.Google Scholar
[15] Moschovakis, Y. N., Descriptive set theory, North-Holland, Amsterdam, 1980.Google Scholar
[16] Puritz, C., Ultrafilters and standard functions in nonstandard arithmetic, Proceedings of the London Mathematical Society, vol. 22 (1971), pp. 705–733.10.1112/plms/s3-22.4.705Google Scholar
[17] Puritz, C., Skies, constellations, and monads, Contributions to nonstandard analysis (Luxemburg, W. A. J. and Robinson, A., editors), North-Holland, Amsterdam, 1972, pp. 215–243.Google Scholar
[18] Sazonov, V. V., On perfect measures, Izvestiya Akademii Nauk SSSR Seriya Matematicheskikh, vol. 26 (1962), pp. 391–414, Translations of the American Mathematical Society , vol. 48, pp. 229–254.Google Scholar
[19] Shelah, S., There may be no nowhere dense ultrafilter, Proceedings of the logic colloquium Haifa '95, Springer-Verlag, to appear.Google Scholar
[20] Shelah, S., Proper forcing, Lecture Notes in Mathematics, no. 940, Springer-Verlag, 1982.Google Scholar
[21] Talagrand, M., Filtres non-measurables et compacts defonctions mesurables, Studia Mathematica, vol. 104 (1980), pp. 13–43.10.4064/sm-67-1-13-43Google Scholar