Some complete Σ¹₂ sets in harmonic analysis

Becker, Howard; Kahane, Sylvain; Louveau, Alain

doi:10.1090/S0002-9947-1993-1129434-8

Some complete $\Sigma ^ 1_ 2$ sets in harmonic analysis
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by Howard Becker, Sylvain Kahane and Alain Louveau

Trans. Amer. Math. Soc. 339 (1993), 323-336

DOI: https://doi.org/10.1090/S0002-9947-1993-1129434-8

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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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