Universal Loeb-measurability of sets and of the standard part map with applications

Landers, D.; Rogge, L.

doi:10.1090/S0002-9947-1987-0906814-1

Universal Loeb-measurability of sets and of the standard part map with applications
HTML articles powered by AMS MathViewer

by D. Landers and L. Rogge PDF

Trans. Amer. Math. Soc. 304 (1987), 229-243 Request permission

Abstract:

It is shown in this paper that for $K$-saturated models many important external sets of nonstandard analysis—such as monadic sets or the set of all near-standard points or all pre-near-standard points or all compact points—are universally Loeb-measurable, i.e. Loeb-measurable with respect to every internal content. We furthermore obtain universal Loeb-measurability of the standard part map for topological spaces which are not covered by previous results in this direction. Moreover, the standard part map can be used as a measure preserving transformation for all $\tau$-smooth measures, and not only for Radon-measures as known up to now. Applications of our results lead to simple new proofs for theorems of classical measure theory. We obtain e.g. the extension of $\tau$-smooth Baire-measures to $\tau$-smooth Borel-measures, the decomposition theorems for $\tau$-smooth Baire-measures and $\tau$-smooth Borel-measures and Kakutani’s theorem for product measures.

References

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
Robert M. Anderson, Star-finite representations of measure spaces, Trans. Amer. Math. Soc. 271 (1982), no. 2, 667–687. MR 654856, DOI 10.1090/S0002-9947-1982-0654856-1
Robert M. Anderson and Salim Rashid, A nonstandard characterization of weak convergence, Proc. Amer. Math. Soc. 69 (1978), no. 2, 327–332. MR 480925, DOI 10.1090/S0002-9939-1978-0480925-X
Nigel J. Cutland, Nonstandard measure theory and its applications, Bull. London Math. Soc. 15 (1983), no. 6, 529–589. MR 720746, DOI 10.1112/blms/15.6.529
C. Ward Henson, Analytic sets, Baire sets and the standard part map, Canadian J. Math. 31 (1979), no. 3, 663–672. MR 536371, DOI 10.4153/CJM-1979-066-0
J. D. Knowles, Measures on topological spaces, Proc. London Math. Soc. (3) 17 (1967), 139–156. MR 204602, DOI 10.1112/plms/s3-17.1.139
Peter A. Loeb, Conversion from nonstandard to standard measure spaces and applications in probability theory, Trans. Amer. Math. Soc. 211 (1975), 113–122. MR 390154, DOI 10.1090/S0002-9947-1975-0390154-8
Peter A. Loeb, Weak limits of measures and the standard part map, Proc. Amer. Math. Soc. 77 (1979), no. 1, 128–135. MR 539645, DOI 10.1090/S0002-9939-1979-0539645-6
Albert T. Bharucha-Reid (ed.), Probabilistic analysis and related topics. Vol. 2, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. MR 556677

Measure spaces in nonstandard models underlying standard stochastic processes

Peter A. Loeb, A functional approach to nonstandard measure theory, Conference in modern analysis and probability (New Haven, Conn., 1982) Contemp. Math., vol. 26, Amer. Math. Soc., Providence, RI, 1984, pp. 251–261. MR 737406, DOI 10.1090/conm/026/737406
W. A. J. Luxemburg, A general theory of monads, Applications of Model Theory to Algebra, Analysis, and Probability (Internat. Sympos., Pasadena, Calif., 1967) Holt, Rinehart and Winston, New York, 1969, pp. 18–86. MR 0244931
A. Sapounakis and M. Sion, On generation of Radon-like measures, Measure theory and its applications (Sherbrooke, Que., 1982) Lecture Notes in Math., vol. 1033, Springer, Berlin, 1983, pp. 283–294. MR 729544, DOI 10.1007/BFb0099866
K. D. Stroyan and W. A. J. Luxemburg, Introduction to the theory of infinitesimals, Pure and Applied Mathematics, No. 72, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976. MR 0491163