Compact measures have Loeb preimages

Author:
David Ross

Journal:
Proc. Amer. Math. Soc. **115** (1992), 365-370

MSC:
Primary 28E05; Secondary 03H05, 28C99, 60B99

MathSciNet review:
1079898

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Abstract: A compact measure is a (possibly nontopological) measure that is inner-regular with respect to a compact family of measurable sets. The main result of this paper is that every compact probability measure is the image, under a measure-preserving transformation, of a Loeb probability space. This generalizes a well-known result about Radon topological probability measures. It is also proved that a compact probability space can be topologized in such a way that the measure is essentially Radon.

**[1]**Sergio Albeverio, Raphael Høegh-Krohn, Jens Erik Fenstad, and Tom Lindstrøm,*Nonstandard methods in stochastic analysis and mathematical physics*, Pure and Applied Mathematics, vol. 122, Academic Press, Inc., Orlando, FL, 1986. MR**859372****[2]**Robert M. Anderson,*Star-finite representations of measure spaces*, Trans. Amer. Math. Soc.**271**(1982), no. 2, 667–687. MR**654856**, 10.1090/S0002-9947-1982-0654856-1**[3]**Nigel J. Cutland,*Nonstandard measure theory and its applications*, Bull. London Math. Soc.**15**(1983), no. 6, 529–589. MR**720746**, 10.1112/blms/15.6.529**[4]**D. Landers and L. Rogge,*Universal Loeb-measurability of sets and of the standard part map with applications*, Trans. Amer. Math. Soc.**304**(1987), no. 1, 229–243. MR**906814**, 10.1090/S0002-9947-1987-0906814-1**[5]**T. Lindstrøm,*Pushing down Loeb measures*, 1981, Preprint Series, Matematisk Institutt, Univ. i Oslo, Mathematics No. 2, March 10.**[6]**Peter A. Loeb,*An introduction to nonstandard analysis and hyperfinite probability theory*, Probabilistic analysis and related topics, Vol. 2, Academic Press, New York-London, 1979, pp. 105–142. MR**556680****[7]**T. Norberg, private correspondence.**[8]**-,*Existence theorems for measures on continuous posets, with applications to random set theory*, Report 1987-11, Dept. of Math., Univ. of Göteborg, 1987.**[9]**M. M. Rao,*Projective limits of probability spaces*, J. Multivariate Anal.**1**(1971), no. 1, 28–57. MR**0301771****[10]**D. Ross,*Measurable transformations in saturated models of analysis*, Ph. D. Thesis, Univ. of Wisconsin, 1983.**[11]**Wim Vervaat,*Random upper semicontinuous functions and extremal processes*, Probability and lattices, CWI Tract, vol. 110, Math. Centrum, Centrum Wisk. Inform., Amsterdam, 1997, pp. 1–56. MR**1465481**

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9939-1992-1079898-8

Keywords:
Loeb measure,
compact measure,
Radon measure

Article copyright:
© Copyright 1992
American Mathematical Society