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Transactions of the American Mathematical Society

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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
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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.

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Article copyright: © Copyright 1980 American Mathematical Society