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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles 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 1.48.

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A separation theorem for $\Sigma ^{1}_{1}$ sets
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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.
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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