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**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)

### MSC:

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 |

### Keywords:

singular measures; Dirichlet measures on the unit circle; non Borel analytic set; weak* Borel set### References:

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