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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Some complete $ \Sigma\sp 1\sb 2$ sets in harmonic analysis


Authors: Howard Becker, Sylvain Kahane and Alain Louveau
Journal: Trans. Amer. Math. Soc. 339 (1993), 323-336
MSC: Primary 04A15; Secondary 03E35, 43A46
MathSciNet review: 1129434
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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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DOI: http://dx.doi.org/10.1090/S0002-9947-1993-1129434-8
PII: S 0002-9947(1993)1129434-8
Article copyright: © Copyright 1993 American Mathematical Society