A simple Efimov space with sequentially-nice space of probability measures

T. Banakh, S. Gabriyelyan

Research output: Contribution to journalArticlepeer-review

Abstract

Under Jensen’s diamond principle ♢ , we construct a simple Efimov space K whose space of nonatomic probability measures Pna(K) is first-countable and sequentially compact. These two properties of Pna(K) imply that the space of probability measures P(K) on K is selectively sequentially pseudocompact. We show that any sequence of probability measures on K that converges to a purely atomic measure converges in norm, and any sequence of probability measures on K converging to zero in sup-norm has a subsequence converging to a nonatomic probability measure. We show also that the Banach space C(K) of continuous functions on K has the Gelfand–Phillips property but it does not have the Grothendieck property.

Original languageEnglish
Article number32
JournalRevista de la Real Academia de Ciencias Exactas, Fisicas y Naturales - Serie A: Matematicas
Volume118
Issue number1
DOIs
StatePublished - 1 Jan 2024

Keywords

  • Efimov space
  • Gelfand–Phillips property
  • Grothendieck property
  • Nonatomic probability measure
  • Selective sequential pseudocompactness
  • Space of probability measures

ASJC Scopus subject areas

  • Analysis
  • Algebra and Number Theory
  • Geometry and Topology
  • Computational Mathematics
  • Applied Mathematics

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