On the complexity of sums of Dirichlet measures
Annales de l'Institut Fourier, Tome 43 (1993) no. 1, pp. 111-123.

Soit M l’ensemble des mesures de Dirichlet sur le tore T. On montre que M+M est un analytique non borélien pour la topologie préfaible et M+M n’est pas fermé en norme. Plus précisément, on montre que M+M ne peut pas être séparé par un borélien préfaible de D⊥ (ou même L0⊥), l’ensemble des mesures étrangères à M, étendant ainsi les résultats de Kaufman, Kechris et Lyons sur D⊥ et H⊥, et exhibant de nombreux exemples d’analytiques non boréliens.

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⊥ (or even L0⊥), the set of all measures singular with respect to every measure in M. This extends results of Kaufman, Kechris and Lyons about D⊥ and H⊥ and gives many examples of non Borel analytic sets.

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     author = {Kahane, Sylvain},
     title = {On the complexity of sums of {Dirichlet} measures},
     journal = {Annales de l'Institut Fourier},
     pages = {111--123},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {43},
     number = {1},
     year = {1993},
     doi = {10.5802/aif.1323},
     mrnumber = {94h:43003},
     zbl = {0766.28001},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/aif.1323/}
}
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Kahane, Sylvain. On the complexity of sums of Dirichlet measures. Annales de l'Institut Fourier, Tome 43 (1993) no. 1, pp. 111-123. doi : 10.5802/aif.1323. https://www.numdam.org/articles/10.5802/aif.1323/

[1] S. Kahane, Antistable classes of thin sets, Illinois J. Math., 37-1 (1993). | MR | Zbl

[2] R. Kaufman, Topics on analytic sets, Fund. Math., 139 (1991), 215-229. | MR | Zbl

[3] A. Kechris, R. Lyons, Ordinal ranking on measures annihilating thin sets, Trans. Amer. Math. Soc., 310 (1988), 747-758. | MR | Zbl

[4] R. Lyons, Mixing and asymptotic distribution modulo 1, Ergod. Th. & Dyn. Syst., 8 (1988), 597-619. | MR | Zbl

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