Some sets obeying harmonic synthesis

Let X be a (not necessarily closed) subspace of the dual space B ^* of a separable Banach space B. Let X ₁ denote the set of all weak ^* limits of sequences in X. Define X _a, for every ordinal number a, by the inductive rule:X _a = (U _{b < a} X _b ) ₁ .There is always a countable ordinal a such that X _a is the weak ^* closure of X; the first such a is called the order of X in B ^* . Let E be a closed subset of a locally compact abelian group. Let PM(E) be the set of pseudomeasures, and M(E) the set of measures, whose supports are contained in E. The set E obeys synthesis if and only if M(E) is weak ^* dense in PM(E). Varopoulos constructed an example in which the order of M(E) is 2. The authors construct, for every countable ordinal a, a set E in R that obeys synthesis, and such that the order of M(E) in PM(E) is a.