Smooth sets for a Borel equivalence relation

Uzcátegui A., Carlos E.

doi:10.1090/S0002-9947-1995-1303127-8

Smooth sets for a Borel equivalence relation
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by Carlos E. Uzcátegui A. PDF

Trans. Amer. Math. Soc. 347 (1995), 2025-2039 Request permission

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.

References

Howard Becker, The restriction of a Borel equivalence relation to a sparse set, Arch. Math. Logic 42 (2003), no. 4, 335–347. MR 2018085, DOI 10.1007/s001530200142
John P. Burgess, A selection theorem for group actions, Pacific J. Math. 80 (1979), no. 2, 333–336. MR 539418, DOI 10.2140/pjm.1979.80.333
R. Dougherty, S. Jackson, and A. S. Kechris, The structure of hyperfinite Borel equivalence relations, Trans. Amer. Math. Soc. 341 (1994), no. 1, 193–225. MR 1149121, DOI 10.1090/S0002-9947-1994-1149121-0
Edward G. Effros, Polish transformation groups and classification problems, General topology and modern analysis (Proc. Conf., Univ. California, Riverside, Calif., 1980) Academic Press, New York-London, 1981, pp. 217–227. MR 619045
Jacob Feldman and Calvin C. Moore, Ergodic equivalence relations, cohomology, and von Neumann algebras. I, Trans. Amer. Math. Soc. 234 (1977), no. 2, 289–324. MR 578656, DOI 10.1090/S0002-9947-1977-0578656-4
L. A. Harrington, A. S. Kechris, and A. Louveau, A Glimm-Effros dichotomy for Borel equivalence relations, J. Amer. Math. Soc. 3 (1990), no. 4, 903–928. MR 1057041, DOI 10.1090/S0894-0347-1990-1057041-5
Alexander S. Kechris, The theory of countable analytical sets, Trans. Amer. Math. Soc. 202 (1975), 259–297. MR 419235, DOI 10.1090/S0002-9947-1975-0419235-7

The descriptive set theory of

-ideals of compact sets

Alexander S. Kechris, Hereditary properties of the class of closed sets of uniqueness for trigonometric series, Israel J. Math. 73 (1991), no. 2, 189–198. MR 1135211, DOI 10.1007/BF02772948
Alexander S. Kechris, Measure and category in effective descriptive set theory, Ann. Math. Logic 5 (1972/73), 337–384. MR 369072, DOI 10.1016/0003-4843(73)90012-0
Alexander S. Kechris, Amenable versus hyperfinite Borel equivalence relations, J. Symbolic Logic 58 (1993), no. 3, 894–907. MR 1242044, DOI 10.2307/2275102
Alexander S. Kechris and Alain Louveau, Descriptive set theory and the structure of sets of uniqueness, London Mathematical Society Lecture Note Series, vol. 128, Cambridge University Press, Cambridge, 1987. MR 953784, DOI 10.1017/CBO9780511758850
A. S. Kechris, A. Louveau, and W. H. Woodin, The structure of $\sigma$-ideals of compact sets, Trans. Amer. Math. Soc. 301 (1987), no. 1, 263–288. MR 879573, DOI 10.1090/S0002-9947-1987-0879573-9
Yiannis N. Moschovakis, Descriptive set theory, Studies in Logic and the Foundations of Mathematics, vol. 100, North-Holland Publishing Co., Amsterdam-New York, 1980. MR 561709
Benjamin Weiss, Measurable dynamics, Conference in modern analysis and probability (New Haven, Conn., 1982) Contemp. Math., vol. 26, Amer. Math. Soc., Providence, RI, 1984, pp. 395–421. MR 737417, DOI 10.1090/conm/026/737417

The covering property for

-ideals of compact sets

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