Hyperfinite transversal theory
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- by Boško Živaljević
- Trans. Amer. Math. Soc. 330 (1992), 371-399
- DOI: https://doi.org/10.1090/S0002-9947-1992-1033237-1
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Abstract:
A measure theoretic version of a well-known P. Hall’s theorem, about the existence of a system of distinct representatives of a finite family of finite sets, has been proved for the case of the Loeb space of an internal, uniformly distributed, hyperfinite counting space. We first prove Hall’s theorem for $\Pi _1^0(\kappa )$ graphs after which we develop the version of discrete Transversal Theory. We then prove a new version of Hall’s theorem in the case of $\Sigma _1^0(\kappa )$ monotone graphs and give an example of a $\Sigma _1^0$ graph which satisfies Hall’s condition and which does not possess an internal a.e. matching.References
- Béla Bollobás, Extremal graph theory, London Mathematical Society Monographs, vol. 11, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1978. MR 506522
- B. Bollobás and N. Th. Varopoulos, Representation of systems of measurable sets, Math. Proc. Cambridge Philos. Soc. 78 (1975), no. 2, 323–325. MR 379781, DOI 10.1017/S0305004100051756 P. Hall, On representatives of subsets, J. London Math. Soc. 10 (1935), 26-30.
- 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 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
- Michael Holz, Klaus-Peter Podewski, and Karsten Steffens, Injective choice functions, Lecture Notes in Mathematics, vol. 1238, Springer-Verlag, Berlin, 1987. MR 880206, DOI 10.1007/BFb0072628
- 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
- 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
- Henry A. Kierstead, An effective version of Hall’s theorem, Proc. Amer. Math. Soc. 88 (1983), no. 1, 124–128. MR 691291, DOI 10.1090/S0002-9939-1983-0691291-0
- 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
- 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
- Alfred B. Manaster and Joseph G. Rosenstein, Effective matchmaking (recursion theoretic aspects of a theorem of Philip Hall), Proc. London Math. Soc. (3) 25 (1972), 615–654. MR 314610, DOI 10.1112/plms/s3-25.4.615
- L. Mirsky, Transversal theory. An account of some aspects of combinatorial mathematics, Mathematics in Science and Engineering, Vol. 75, Academic Press, New York-London, 1971. MR 0282853
- C. St. J. A. Nash-Williams, Unexplored and semi-explored territories in graph theory, New directions in the theory of graphs (Proc. Third Ann Arbor Conf., Univ. Michigan, Ann Arbor, Mich., 1971) Academic Press, New York, 1973, pp. 149–186. MR 0387097 R. Rado, A theorem on general measure function, Proc. London Math. Soc. 44 (1938), 61-91. D. Ross, (private communication).
- K. D. Stroyan and José Manuel Bayod, Foundations of infinitesimal stochastic analysis, Studies in Logic and the Foundations of Mathematics, vol. 119, North-Holland Publishing Co., Amsterdam, 1986. MR 849100
- 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
- Boško Živaljević, Rado’s theorem for the Loeb space of an internal $*$-finitely additive measure space, Proc. Amer. Math. Soc. 112 (1991), no. 1, 203–207. MR 1056688, DOI 10.1090/S0002-9939-1991-1056688-2
- Boško Živaljević, The structure of graphs all of whose $Y$-sections are internal sets, J. Symbolic Logic 56 (1991), no. 1, 50–66. MR 1131729, DOI 10.2307/2274903 —, Hyperfinite transversal theory. II (in preparation).
Bibliographic Information
- © Copyright 1992 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 330 (1992), 371-399
- MSC: Primary 03H05; Secondary 04A20, 05D15
- DOI: https://doi.org/10.1090/S0002-9947-1992-1033237-1
- MathSciNet review: 1033237