Transactions of the American Mathematical Society

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



A separation theorem for $ \Sigma \sp{1}\sb{1}$ sets

Author: Alain Louveau
Journal: Trans. Amer. Math. Soc. 260 (1980), 363-378
MSC: Primary 04A15; Secondary 03E15, 26A21, 28A05, 54H05
MathSciNet review: 574785
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we show that the notion of Borel class is, roughly speaking, an effective notion. We prove that if a set A is both $ \prod _\xi ^0$ and $ \Delta _1^1$, it possesses a $ \Pi _\xi ^0$-code which is also $ \Delta _1^1$. As a by-product of the induction used to prove this result, we also obtain a separation result for $ \Sigma _1^1$ sets: If two $ \Sigma _1^1$ sets can be separated by a $ \Pi _\xi ^0$ set, they can also be separated by a set which is both $ \Delta _1^1$ and $ \Pi _\xi ^0$.

Applications of these results include a study of the effective theory of Borel classes, containing separation and reduction principles, and an effective analog of the Lebesgue-Hausdorff theorem on analytically representable functions. We also give applications to the study of Borel sets and functions with sections of fixed Borel class in product spaces, including a result on the conservation of the Borel class under integration.

References [Enhancements On Off] (What's this?)

  • [Bo1] J. Bourgain, Decompositions in the product of a measure space and a Polish space, Fund. Math. 105 (1979/80), no. 1, 61–71. MR 558129
  • [Bo2] J. Bourgain, 𝐹_{𝜎𝛿}-sections of Borel sets, Fund. Math. 107 (1980), no. 2, 129–133. MR 584665
  • [Bo3] -, Borel sets with $ {F_{\sigma \delta }}$-sections (unpublished).
  • [Bu] J. Burgess, Effective Hausdorff resolution (unpublished).
  • [Ce] Douglas Cenzer, Monotone inductive definitions over the continuum, J. Symbolic Logic 41 (1976), no. 1, 188–198. MR 0427054
  • [De] C. Dellacherie, Ensembles analytiques. Théorèmes de séparation et applications, Séminaire de Probabilités, IX (Seconde Partie, Univ. Strasbourg, Strasbourg, années universitaires 1973/1974 et 1974/1975), Springer, Berlin, 1975, pp. 336–372. Lecture Notes in Math., Vol. 465. MR 0428306
  • [Ha] L. Harrington, A powerless proof of a theorem of Silver (circulated manuscript).
  • [Ke] A. S. Kechris, Course on descriptive set theory (circulated manuscript).
  • [Ku] K. Kuratowski, Topology. Vol. I, New edition, revised and augmented. Translated from the French by J. Jaworowski, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe, Warsaw, 1966. MR 0217751
  • [Lo1] A. Louveau, Recursivity and compactness, Higher set theory (Proc. Conf., Math. Forschungsinst., Oberwolfach, 1977), Lecture Notes in Math., vol. 669, Springer, Berlin, 1978, pp. 303–337. MR 520192
  • [Lo2] -, Boréliens à coupes $ {K_{\sigma \delta }}$, C. R. Acad. Sci. Paris 285 (1977), 309-312.
  • [Lo3] Alain Louveau, La hiérarchie borélienne des ensembles 𝐷¹₁, C. R. Acad. Sci. Paris Sér. A-B 285 (1977), no. 9, A601–A604 (French, with English summary). MR 0446982
  • [Lo4] Alain Louveau, Sur les fonctions boréliennes de plusieurs variables, C. R. Acad. Sci. Paris Sér. A-B 285 (1977), no. 16, A1037–A1039 (French, with English summary). MR 0583547
  • [Mo] 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
  • [StR] Jean Saint-Raymond, Boréliens à coupes 𝐾_{𝜎}, Bull. Soc. Math. France 104 (1976), no. 4, 389–400. MR 0433418
  • [GMS] S. Grigorief, K. Mc Aloon and J. Stern, Séminaire de théorie des ensembles, 1976-1977, Publ. de Paris VII.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 04A15, 03E15, 26A21, 28A05, 54H05

Retrieve articles in all journals with MSC: 04A15, 03E15, 26A21, 28A05, 54H05

Additional Information

Article copyright: © Copyright 1980 American Mathematical Society