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

T. Banakh; S. Gabriyelyan

doi:10.1007/s13398-023-01531-6

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

T. Banakh, S. Gabriyelyan

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

Abstract

Under Jensen’s diamond principle ♢ , we construct a simple Efimov space K whose space of nonatomic probability measures P_na(K) is first-countable and sequentially compact. These two properties of P_na(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 language	English
Article number	32
Journal	Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales - Serie A: Matematicas
Volume	118
Issue number	1
DOIs	https://doi.org/10.1007/s13398-023-01531-6
State	Published - 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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10.1007/s13398-023-01531-6

Cite this

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abstract = "Under Jensen{\textquoteright}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.",

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

Abstract

Keywords

ASJC Scopus subject areas

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