On sequences of finitely supported measures related to the Josefson–Nissenzweig theorem

Abstract

Given a Tychonoff space X, we call a sequence 〈μn:n∈ω〉 of signed Borel measures on X a finitely supported Josefson–Nissenzweig sequence (in short a JN-sequence) if: 1) for every n∈ω the measure μn is a finite combination of one-point measures and ‖μn‖=1, and 2) ∫Xfdμn→0 for every continuous function f∈C(X). Our main result asserts that if a Tychonoff space X admits a JN-sequence, then there exists a JN-sequence 〈μn:n∈ω〉 such that: i) supp(μn)∩supp(μk)=∅ for every n≠k∈ω, and ii) the union ⋃n∈ωsupp(μn) is a discrete subset of X. We also prove that if a Tychonoff space X carries a JN-sequence, then either there is a JN-sequence 〈μn:n∈ω〉 on X such that |supp(μn)|=2 for every n∈ω, or for every JN-sequence 〈μn:n∈ω〉 on X we have limn→∞⁡|supp(μn)|=∞.