The class of synthesizable pseudomeasures
Alexander S. Kechris, Alain Louveau, V. Tardivel · Illinois Journal of Mathematics · 1991
In this paper we study descriptive set theoretic questions related to concepts of harmonic synthesis on the unit circle T, and their relationship with the structure of uniqueness sets. We denote by A A(T) the space of functions on T with absolutely convergent Fourier series, by PM the space of pseudomeasures on T and by PF the space of pseudo-functions on T.Thus PF* A, A* PM.Finally K(T) denotes the compact space of closed subsets of T with the Hausdorff metric.The three basic notions associated with harmonic synthesis are the following: (i) A function f A satisfies synthesis if (f, S) 0 for all S PM with f 0 on supp(S).(ii) A pseudomeasure S PM satisfies synthesis if (f, S)= 0 for all fA with f= 0 on supp(S).This is equivalent to saying that S N(supp(S)), where for each E K(T), we let M(E) space of (Borel complex) measures whose (closed) support is contained in E, N(E) weak *-closure of M(E).For simplicity, if S PM satisfies synthesis, we will call it a synthesizable pseudomeasure.(iii) A set E K(T) is a set of synthesis if for all f A, S PM with supp(S) _ _ _ E and f 0 on E we have (f, S) 0. Equivalently, if I(E) {f A" f= 0on E}, J(E) {f A" f 0 on an (open) nbhd of E}, E is of synthesis iff the strong closure of J(E) in A is equal to I(E). Also equivalently, E is of synthesis iff N(E)= PM(E) (= the space of pseu- domeasures supported by E).