On the complexity of sums of Dirichlet measures. (English) Zbl 0766.28001

Let \(M\) be the set of all Dirichlet measures on the unit circle. We prove that \(M+M\) is a non Borel analytic set for the weak* topology and that \(M+M\) is not norm-closed. More precisely, we prove that there is no weak* Borel set which separates \(M+M\) from \(D^ \perp\) (or even \(L^ \perp_ 0)\), the set of all measures singular with respect to every measure in \(M\). This extends results of Kaufman, Kechris and Lyons about \(D^ \perp\) and \(H^ \perp\) and gives many examples of non Borel analytic sets.
Reviewer: S.Kahane (Paris)


28A33 Spaces of measures, convergence of measures
28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
28A12 Contents, measures, outer measures, capacities
03E15 Descriptive set theory
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[1] [1] , Antistable classes of thin sets, Illinois J. Math., 37-1 (1993). · Zbl 0793.42003
[2] [2] , Topics on analytic sets, Fund. Math., 139 (1991), 215-229. · Zbl 0764.28002
[3] [3] , , Ordinal ranking on measures annihilating thin sets, Trans. Amer. Math. Soc., 310 (1988), 747-758. · Zbl 0706.43007
[4] [4] , Mixing and asymptotic distribution modulo 1, Ergod. Th. & Dyn. Syst., 8 (1988), 597-619. · Zbl 0645.10042
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