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

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 L 0 ), 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.

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 L 0 ), 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.

@article{AIF_1993__43_1_111_0,
     author = {Kahane, Sylvain},
     title = {On the complexity of sums of Dirichlet measures},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Louis-Jean},
     address = {Gap},
     volume = {43},
     number = {1},
     year = {1993},
     pages = {111-123},
     doi = {10.5802/aif.1323},
     zbl = {0766.28001},
     mrnumber = {94h:43003},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1993__43_1_111_0}
}
Kahane, Sylvain. On the complexity of sums of Dirichlet measures. Annales de l'Institut Fourier, Volume 43 (1993) no. 1, pp. 111-123. doi : 10.5802/aif.1323. http://www.numdam.org/item/AIF_1993__43_1_111_0/

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