A Loeb-measure approach to theorems by Prohorov, Sazonov and Gross
HTML articles powered by AMS MathViewer
- by Tom L. Lindstrøm PDF
- Trans. Amer. Math. Soc. 269 (1982), 521-534 Request permission
Abstract:
We use the Loeb-measure of nonstandard analysis to prove three classical results on limit measures: Let ${\{ {\mu _i}\} _{i \in I}}$ be a projective system of Radon measures, we use the Loeb-measure $L({\tilde \mu _E})$ for an infinite $E \in {}^{\ast }I$ and a standard part map to construct a Radon limit measure on the projective limit (Prohorov’s Theorem). Using the Loeb-measures on hyperfinite dimensional linear spaces, we characterize the Fourier-transforms of measures on Hilbert spaces (Sazonov’s Theorem), and extend cylindrical measures on Hilbert spaces to $\sigma$-additive measures on Banach spaces (Gross’ Theorem).References
- Robert M. Anderson, A non-standard representation for Brownian motion and Itô integration, Israel J. Math. 25 (1976), no. 1-2, 15–46. MR 464380, DOI 10.1007/BF02756559 —, Star-finite probability theory, Ph.D. thesis, Yale University, 1977.
- 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
- Patrick Billingsley, Convergence of probability measures, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0233396
- Leonard Gross, Abstract Wiener spaces, Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66) Univ. California Press, Berkeley, Calif., 1967, pp. 31–42. MR 0212152 L. L. Helms, A nonstandard approach to the martingale problem for spin models, University of Illinois, 1979 (preprint).
- L. L. Helms and P. A. Loeb, Applications of nonstandard analysis to spin models, J. Math. Anal. Appl. 69 (1979), no. 2, 341–352. MR 538222, DOI 10.1016/0022-247X(79)90147-1
- 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
- A. E. Hurd, Nonstandard analysis and lattice statistical mechanics: a variational principle, Trans. Amer. Math. Soc. 263 (1981), no. 1, 89–110. MR 590413, DOI 10.1090/S0002-9947-1981-0590413-2
- Hui Hsiung Kuo, Gaussian measures in Banach spaces, Lecture Notes in Mathematics, Vol. 463, Springer-Verlag, Berlin-New York, 1975. MR 0461643
- 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 —, An introduction to non-standard analysis and hyperfinite probability theory, Probabilistic Analysis and Related Topics 2 (A. T. Bharucha-Reid, editor), Academic Press, New York, 1979, pp. 105-142.
- Yu. V. Prokhorov, Convergence of random processes and limit theorems in probability theory, Teor. Veroyatnost. i Primenen. 1 (1956), 177–238 (Russian, with English summary). MR 0084896
- Michael Reed and Barry Simon, Methods of modern mathematical physics. I, 2nd ed., Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1980. Functional analysis. MR 751959
- V. Sazonov, On characteristic functionals, Teor. Veroyatnost. i Primenen. 3 (1958), 201–205 (Russian, with English summary). MR 0098423
- Laurent Schwartz, Radon measures on arbitrary topological spaces and cylindrical measures, Tata Institute of Fundamental Research Studies in Mathematics, No. 6, Published for the Tata Institute of Fundamental Research, Bombay by Oxford University Press, London, 1973. MR 0426084
- 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
- N. Bourbaki, Éléments de mathématique. Fasc. XX. Livre I: Théorie des ensembles. Chapitre 3: Ensembles ordonnés cardinaux, nombres entiers, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1243, Hermann, Paris, 1963 (French). Seconde édition, revue et augmentée. MR 0154814 —, Topologie générale, Chapitre I, 4th ed., Hermann, Paris, 1965.
- D. W. Müller, Nonstandard proofs of invariance principles in probability theory, Applications of Model Theory to Algebra, Analysis, and Probability (Internat. Sympos., Pasadena, Calif., 1967) Holt, Rinehart and Winston, New York, 1969, pp. 186–194. MR 0239645
Additional Information
- © Copyright 1982 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 269 (1982), 521-534
- MSC: Primary 60B11; Secondary 03H05, 26E35, 28C20
- DOI: https://doi.org/10.1090/S0002-9947-1982-0637706-9
- MathSciNet review: 637706