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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Smooth sets for a Borel equivalence relation

Author: Carlos E. Uzcátegui A.
Journal: Trans. Amer. Math. Soc. 347 (1995), 2025-2039
MSC: Primary 03E15; Secondary 04A15, 28A05, 28D99, 54H05, 54H20
MathSciNet review: 1303127
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Abstract: We study some properties of smooth Borel sets with respect to a Borel equivalence relation, showing some analogies with the collection of countable sets from a descriptive set theoretic point of view. We found what can be seen as an analog of the hyperarithmetic points in the context of smooth sets. We generalize a theorem of Weiss from $ {\mathbf{Z}}$-actions to actions by arbitrary countable groups. We show that the $ \sigma $-ideal of closed smooth sets is $ \Pi _1^1$ non-Borel.

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  • [1] H. Becker, The restriction of a Borel equivalence relation to a sparse set, preprint, 1992. MR 2018085 (2004h:03099)
  • [2] J. Burgess, A selection theorem for group actions, Pacific J. Math. 80 (1979), 333-336. MR 539418 (81a:54041)
  • [3] R. Dougherty, S. Jackson and A. S. Kechris, The structure of hyperfinite Borel equivalence relations, Trans. Amer. Math. Soc. 341 (1994), 193-225. MR 1149121 (94c:03066)
  • [4] E. Effros, Polish transformation groups and classification problems, General Topology and Modern Analysis, (M. M. Rao and L. F. Mc Auley, eds.), Academic Press, 1980, pp. 217-227. MR 619045 (82k:54064)
  • [5] J. Feldman and C. C. Moore, Ergodic equivalence relations, cohomology and von Neumann algebras, I, Trans. Amer. Math. Soc. 234 (1977), 289-324. MR 0578656 (58:28261a)
  • [6] L. A. Harrington, A. S. Kechris and A. Louveau, A Glimm-Effros dichotomy for Borel equivalence relations, J. Amer. Math. Soc. 3 (1990), 903-928. MR 1057041 (91h:28023)
  • [7] A. S. Kechris, The theory of countable analytical sets, Trans. Amer. Math. Soc. 202 (1975), 259-297. MR 0419235 (54:7259)
  • [8] -, The descriptive set theory of $ \sigma $-ideals of compact sets, Logic Colloquium'88, (R. Ferro, C. Bonotto, S. Valentini and A. Zanardo, eds.), North-Holland, 1989, pp. 117-138.
  • [9] -, Hereditary properties of the class of closed uniqueness sets, Israel J. Math. 73 (1991), 189-198. MR 1135211 (93c:42008)
  • [10] -, Measure and category in effective descriptive set theory, Ann. Math. Logic 5 (1973), 337-384. MR 0369072 (51:5308)
  • [11] -, Amenable versus hyperfinite Borel equivalence relations, J. Symbolic Logic 58 (1993), 894-907. MR 1242044 (95f:03081)
  • [12] A. S. Kechris and A. Louveau, Descriptive set theory and the structure of sets of uniqueness, LMS Lecture Notes, 128, Cambridge Univ. Press, 1987. MR 953784 (90a:42008)
  • [13] A.S. Kechris, Lectures on Borel equivalence relation, unpublished.
  • [14] A. S. Kechris, A. Louveau and W. H. Woodin, The structure of $ \sigma $-ideals of compact sets, Trans. Amer. Math. Soc. 301 (1987), 263-288. MR 879573 (88f:03042)
  • [15] Y. N. Moschovakis, Descriptive set theory, North-Holland, Amsterdam, 1980. MR 561709 (82e:03002)
  • [16] B. Weiss, Measurable dynamics, Contemp. Math. 26 (1984), 395-421. MR 737417 (85j:28027)
  • [17] C. Uzcátegui, Ph.D. Thesis, Caltech, 1990.
  • [18] -, The covering property for $ \sigma $-ideals of compact sets, Fund. Math. 140 (1992), 119-146.

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Additional Information

Keywords: Borel equivalence relations, negligible sets, $ \sigma $-ideals of compact sets, group actions, wandering sets
Article copyright: © Copyright 1995 American Mathematical Society

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