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Hereditary properties of the class of closed sets of uniqueness for trigonometric series. (English) Zbl 0769.42006

Summary: It is shown that the \(\sigma\)-ideal \(U_ 0\) of closed sets of extended uniqueness in \(\mathbb{T}\), \(\mathbb{T}=\mathbb{R}/2\pi\mathbb{Z}\) the circle group, is hereditarily non-Borel, i.e., every “nontrivial” \(\sigma\)-ideal of closed sets \(I\subseteq U_ 0\) is non-Borel. This implies both the result of Solovay and Kaufman that both \(U_ 0\) and \(U\) (the \(\sigma\)- ideal of closed sets of uniqueness) are not Borel as well as the theorem of Debs-Saint Raymond that every Borel subset of \(\mathbb{T}\) of extended uniqueness is of the first category. A further extension to ideals contained in \(U_ 0\) is given.

MSC:

42A63 Uniqueness of trigonometric expansions, uniqueness of Fourier expansions, Riemann theory, localization
03E15 Descriptive set theory
28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
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References:

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