Sequential closure in the space of measures

Abstract

We show that there is a compact topological space carrying a measure which is not a weak* limit of finitely supported measures but is in the sequential closure of the set of such measures. We construct compact spaces with measures of arbitrarily high levels of complexity in this sequential hierarchy. It follows that there is a compact space in which the sequential closure cannot be obtained in countably many steps. However, we show that this is not the case for our spaces where the sequential closure is always obtained in countably many steps.Comment: (18 pages, a gap in an argument from the previous version fixed

On Baire measurability in spaces of continuous functions

“…In fact, the compact space K of Theorem 2.7 satisfies Seq 1 (co∆ K ) = Seq 2 (co∆ K ) and Seq 3 (co∆ K ) = P (K). Along this way, it was recently proven in [3] (without additional set-theoretic assumptions) that for every ordinal 1 ≤ α < ω 1 there is a compact space K (α) such that Seq α (co∆ K (α) ) \ β<α Seq β (co∆ K (α) ) = ∅ and Seq α+1 (co∆ K (α) ) = Seq(co∆ K (α) ) = P (K (α) ).…”

“…In Section 3 we pay further attention to (1.2) and show that it fails for K = βω and K = βω \ ω (Theorem 3.4 and Corollary 3. 3). Some related open problems are posed at the end of the paper.…”