Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


A separation theorem for $\Sigma ^{1}_{1}$ sets
HTML articles powered by AMS MathViewer

by Alain Louveau PDF
Trans. Amer. Math. Soc. 260 (1980), 363-378 Request permission


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.
  • 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, DOI 10.4064/fm-105-1-61-71
  • J. Bourgain, $F_{\sigma \delta }$-sections of Borel sets, Fund. Math. 107 (1980), no. 2, 129–133. MR 584665, DOI 10.4064/fm-107-2-129-133
  • —, Borel sets with ${F_{\sigma \delta }}$-sections (unpublished). J. Burgess, Effective Hausdorff resolution (unpublished).
  • Douglas Cenzer, Monotone inductive definitions over the continuum, J. Symbolic Logic 41 (1976), no. 1, 188–198. MR 427054, DOI 10.2307/2272958
  • 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), Lecture Notes in Math., Vol. 465, Springer, Berlin, 1975, pp. 336–372. MR 0428306
  • L. Harrington, A powerless proof of a theorem of Silver (circulated manuscript). A. S. Kechris, Course on descriptive set theory (circulated manuscript).
  • K. Kuratowski, Topology. Vol. I, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw, 1966. New edition, revised and augmented; Translated from the French by J. Jaworowski. MR 0217751
  • 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
  • —, Boréliens à coupes ${K_{\sigma \delta }}$, C. R. Acad. Sci. Paris 285 (1977), 309-312.
  • Alain Louveau, La hiérarchie borélienne des ensembles $D^{1}_{1}$, C. R. Acad. Sci. Paris Sér. A-B 285 (1977), no. 9, A601–A604 (French, with English summary). MR 446982
  • 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 583547
  • 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
  • Jean Saint-Raymond, Boréliens à coupes $K_{\sigma }$, Bull. Soc. Math. France 104 (1976), no. 4, 389–400. MR 433418, DOI 10.24033/bsmf.1835
  • S. Grigorief, K. Mc Aloon and J. Stern, Séminaire de théorie des ensembles, 1976-1977, Publ. de Paris VII.
Similar Articles
Additional Information
  • © Copyright 1980 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 260 (1980), 363-378
  • MSC: Primary 04A15; Secondary 03E15, 26A21, 28A05, 54H05
  • DOI:
  • MathSciNet review: 574785