On the complexity of sums of Dirichlet measures (Q1210518)

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.