Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-10T16:28:17.119Z Has data issue: false hasContentIssue false
  • English
  • Français

A Characterization of Universal Loeb Measurability for Completely Regular Hausdorff Spaces

Published online by Cambridge University Press:  20 November 2018

J. M. Aldaz
J. M. Aldaz*
Affiliation:
Department of Mathematics, The University of the West Indies, Kingston7, Jamaica
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper it is shown that the construction of measures on standard spaces via Loeb measures and the standard part map does not depend on the full structure of the internal algebra being used. A characterization of universal Loeb measurability is given for completely regular Hausdorff spaces, and the behavior of this property under various topological operations is investigated.

Keywords

Primary: 03H05secondary: 28A12
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 44 , Issue 4 , 01 August 1992 , pp. 673 - 690
Copyright
Copyright © Canadian Mathematical Society 1992

References

1. Albeverio, S., Fenstad, J.E., Hoegh-Krohn, R., and Lindstr, T.0m, Nonstandard methods in stochastic analysis and mathematical physics, Academic Press, New York, 1986.Google Scholar
2. Anderson, R.M., Star-finite representations of measure spaces, Trans. Amer. Math. Soc. 271 (1982), 667- 687.Google Scholar
3. Engelking, R., General Topology, Berlin: Heldermann, 1989.Google Scholar
4. H, D.J.. Garling, Another ‘short’ proof of the Riesz representation theorem, Math. Proc. Cambridge Philos. Soc. 99 (1986), 261–262.Google Scholar
5. Gardner, R.J. and Pfeffer, W.F., Borel measures, Handbook of Set Theoretic Topology, (Kunen, K. and Vaughan, J.E., eds.), North-Holland, Amsterdam, 1984.961–1043.Google Scholar
6. Henson, C.W., Analytic sets, Baire sets and the standard part map, Canad. J. Math. 31 (1979), 663–672.Google Scholar
7. Horn, A. and Tarski, A., Measures in Boolean algebras, Trans. Amer. Math. Soc. 64 (1948), 467–497.Google Scholar
8. Knowles, J.D., Measures on topological spaces, Proc. London Math. Soc. (3) 17 (1967), 139–156.Google Scholar
9. Lindstr, T.öm, An invitation to Nonstandard Analysis. In: Nonstandard Analysis and its applications, (Cutland, N. éd.), London Mathematical Society Student Text 10, Cambridge University Press, 1988.Google Scholar
10. Loeb, P.A., Conversion from nonstandard to standard measure spaces and applications in probability theory, Trans. Amer. Math. Soc. 211 (1975), 113–122.Google Scholar
11. Loeb, P.A., Weak limits of measures and the standard part map, Proc. Amer. Math. Soc. 77 (1979), 128–135.Google Scholar
12. Loeb, P.A., Afunctional approach to nonstandard measure theory, Conference on Modern Analysis and Probability Theory, (Beals et. al. eds.), Amer. Math. Soc, Providence, R.I., 1984.Google Scholar
13. Loeb, P.A., Applications of nonstandard analysis to ideal boundaries in potential theory, Israel J. Math. 25 (1976), 154–187.Google Scholar
14. Landers, D. and Rogge, L., Universal Loeb-measurability of sets and of the standard part map with applications, Trans. Amer. Math. Soc. 304 (1987), 229–243.Google Scholar
15. J, W.A.. Luxemburg, A general theory of monads, Applications of Model Theory to Algebra, Analysis and Probability, (J, W.A.. Luxemburg, éd.), Holt, Rinehart and Winston, New York, 1969.18–69.Google Scholar
16. Ross, D., Yet another short proof of the Riesz representation theorem, Math. Proc. Cambridge Philos. Soc. 105 (1989), 261–262.Google Scholar
17. Ross, D., Lifting theorems in nonstandard measure theory, Proc. Amer. Math. Soc, to appear.Google Scholar
18. Stroyan, K.D. and Bayod, J.M., Foundations of Infinitesimal Stochastic Analysis, North-Holland, Amsterdam, New York, Oxford, Tokyo, 1986.Google Scholar
19. Wheeler, R.F., A survey of Baire measures and strict topologies, Expositiones Math. 2 (1983), 97–190.Google Scholar
20. Zivaljevic, R., A Loeb measure approach to the Riesz representation theorem, Pub. Inst. Math., Beograd, N.S. 32 (1982), 175–177.Google Scholar