2023/03/07 by Witold Marciszewski, Marciszewski, Witold, Damian Sobota +3 · 1 citation
Mathematics · #Advanced Banach Space Theory #Advanced Topology and Set Theory #FOS: Mathematics #Functional Analysis (math.FA) #General Topology (math.GN)
paper · pdf · doi:10.48550/arxiv.2303.03809
openalex publication_date 2023/03/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a Tychonoff space X, we call a sequence ⟨μn\colon 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) ∫Xf dμn→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\colon n∈ω⟩ such that: i) supp(μn)\capsupp(μk)=∅ for every n≠ k∈ω, and ii) the union \bigcupn∈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\colon n∈ω⟩ on X such that |supp(μn)|=2 for every n∈ω, or for every JN-sequence ⟨μn\colon n∈ω⟩ on X we have limn→∞|supp(μn)|=∞.