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In [2], D. E. Grow and M. Insall construct a countable compact set which is not the union of two H-sets. We make precise this result in two directions, proving such a set may be, but need not be, a finite union of H-sets. Descriptive set theory tools like Cantor-Bendixson ranks are used; they are developed in the book of A. S. Kechris and A. Louveau [6]. Two proofs are presented; the first one is elementary while the second one is more general and useful. Using the last one I prove in my thesis,...

In this paper, motivated by questions in Harmonic Analysis, we study the operation of (countable) increasing union, and show it is not idempotent: ${\omega }_1$ iterations are needed in general to obtain the closure of a class under this operation. Increasing union is a particular Hausdorff operation, and we present the combinatorial tools which allow to study the power of various Hausdorff operations, and of their iterates. Besides countable increasing union, we study in detail a related Hausdorff operation,...

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 $L_0^{\perp })$, 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.