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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.

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Some complete $\Sigma ^ 1_ 2$ sets in harmonic analysis
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by Howard Becker, Sylvain Kahane and Alain Louveau PDF
Trans. Amer. Math. Soc. 339 (1993), 323-336 Request permission

Abstract:

We prove that several specific pointsets are complete $\Sigma _2^1$ (complete PCA). For example, the class of ${N_0}$-sets, which is a hereditary class of thin sets that occurs in harmonic analysis, is a pointset in the space of compact subsets of the unit circle; we prove that this pointset is complete $\Sigma _2^1$. We also consider some other aspects of descriptive set theory, such as the nonexistence of Borel (and consistently with ${\text {ZFC}}$, the nonexistence of universally measurable) uniformizing functions for several specific relations. For example, there is no Borel way (and consistently, no measurable way) to choose for each ${N_0}$-set, a trigonometric series witnessing that it is an ${N_0}$-set.
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Additional Information
  • © Copyright 1993 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 339 (1993), 323-336
  • MSC: Primary 04A15; Secondary 03E35, 43A46
  • DOI: https://doi.org/10.1090/S0002-9947-1993-1129434-8
  • MathSciNet review: 1129434