Uniqueness of unbounded Loeb measure using Choquet’s theorem
HTML articles powered by AMS MathViewer
- by Boško Živaljević PDF
- Proc. Amer. Math. Soc. 116 (1992), 529-533 Request permission
Abstract:
Uniqueness of the Carathéodory extension of the standard part map of an internal unbounded measure $\mu$ defined on an internal algebra $\mathcal {A}$ of subsets of an internal set $\Omega$ has been proved by Henson using the notion of a countably determined set. Here we show how Choquet’s capacitability theorem can be used in the proof of the same result.References
- D. W. Bressler and M. Sion, The current theory of analytic sets, Canadian J. Math. 16 (1964), 207–230. MR 163854, DOI 10.4153/CJM-1964-021-7
- Gustave Choquet, Theory of capacities, Ann. Inst. Fourier (Grenoble) 5 (1953/54), 131–295 (1955). MR 80760, DOI 10.5802/aif.53
- 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
- C. Ward Henson, Unbounded Loeb measures, Proc. Amer. Math. Soc. 74 (1979), no. 1, 143–150. MR 521888, DOI 10.1090/S0002-9939-1979-0521888-9
- C. Ward Henson and David Ross, Analytic mappings on hyperfinite sets, Proc. Amer. Math. Soc. 118 (1993), no. 2, 587–596. MR 1126195, DOI 10.1090/S0002-9939-1993-1126195-9
- Albert E. Hurd and Peter A. Loeb, An introduction to nonstandard real analysis, Pure and Applied Mathematics, vol. 118, Academic Press, Inc., Orlando, FL, 1985. MR 806135
- Analytic sets, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1980. Lectures delivered at a Conference held at University College, University of London, London, July 16–29, 1978. MR 608794
- H. Jerome Keisler, Kenneth Kunen, Arnold Miller, and Steven Leth, Descriptive set theory over hyperfinite sets, J. Symbolic Logic 54 (1989), no. 4, 1167–1180. MR 1026596, DOI 10.2307/2274812
- 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
- David Ross, Lifting theorems in nonstandard measure theory, Proc. Amer. Math. Soc. 109 (1990), no. 3, 809–822. MR 1019753, DOI 10.1090/S0002-9939-1990-1019753-0
- Boško Živaljević, Every Borel function is monotone Borel, Ann. Pure Appl. Logic 54 (1991), no. 1, 87–99. MR 1130220, DOI 10.1016/0168-0072(91)90010-J
- Boško Živaljević, Some results about Borel sets in descriptive set theory of hyperfinite sets, J. Symbolic Logic 55 (1990), no. 2, 604–614. MR 1056374, DOI 10.2307/2274650
- Boško Živaljević, $U$-meager sets when the cofinality and the coinitiality of $U$ are uncountable, J. Symbolic Logic 56 (1991), no. 3, 906–914. MR 1129155, DOI 10.2307/2275060
Additional Information
- © Copyright 1992 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 116 (1992), 529-533
- MSC: Primary 28E05; Secondary 03H05
- DOI: https://doi.org/10.1090/S0002-9939-1992-1094509-3
- MathSciNet review: 1094509