Representing abstract measures by Loeb measures: a generalization of the standard part map
HTML articles powered by AMS MathViewer
- by J. M. Aldaz
- Proc. Amer. Math. Soc. 123 (1995), 2799-2808
- DOI: https://doi.org/10.1090/S0002-9939-1995-1260159-1
- PDF | Request permission
Abstract:
Loeb measures have been utilized to represent Radon and $\tau$-smooth measures on topological spaces via the standard part map. The purpose of this paper is to show how to extend these results to a nontopological setting.References
- 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
- Allen R. Bernstein and Frank Wattenberg, Nonstandard measure theory, Applications of Model Theory to Algebra, Analysis, and Probability (Internat. Sympos., Pasadena, Calif., 1967) Holt, Rinehart and Winston, New York, 1969, pp. 171–185. MR 0247018
- C. Ward Henson, On the nonstandard representation of measures, Trans. Amer. Math. Soc. 172 (1972), 437–446. MR 315082, DOI 10.1090/S0002-9947-1972-0315082-2
- 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, Applications of nonstandard analysis to ideal boundaries in potential theory, Israel J. Math. 25 (1976), no. 1-2, 154–187. MR 457757, DOI 10.1007/BF02756567 —, A nonstandard representation of measurable spaces, ${L_\infty }$, and $L_\infty ^ \ast$, Contributions to Nonstandard Analysis (W. A. J. Luxemburg and A. Robinson, eds.), North-Holland, Amsterdam, 1972, pp. 65-80.
- 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, DOI 10.1090/S0002-9947-1987-0906814-1
- W. A. J. Luxemburg (ed.), Applications of model theory to algebra, analysis, and probability. , Holt, Rinehart and Winston, New York-Montreal, Que.-London, 1969. MR 0234829
- David Ross, Compact measures have Loeb preimages, Proc. Amer. Math. Soc. 115 (1992), no. 2, 365–370. MR 1079898, DOI 10.1090/S0002-9939-1992-1079898-8
- Flemming Topsøe, Topology and measure, Lecture Notes in Mathematics, Vol. 133, Springer-Verlag, Berlin-New York, 1970. MR 0422560, DOI 10.1007/BFb0069481
Bibliographic Information
- © Copyright 1995 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 123 (1995), 2799-2808
- MSC: Primary 28E05; Secondary 03H05, 28A12
- DOI: https://doi.org/10.1090/S0002-9939-1995-1260159-1
- MathSciNet review: 1260159