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

Ben-Gurion University of the Negev

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

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	American 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

All Science Journal Classification (ASJC) codes

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

Access to Document

10.1007/s13398-023-01531-6

Cite this

@article{408d4599460042a69726497319822c55,

title = "A simple Efimov space with sequentially-nice space of probability measures",

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.",

keywords = "Efimov space, Gelfand–Phillips property, Grothendieck property, Nonatomic probability measure, Selective sequential pseudocompactness, Space of probability measures",

author = "T. Banakh and S. Gabriyelyan",

note = "Publisher Copyright: {\textcopyright} 2023, The Author(s) under exclusive licence to The Royal Academy of Sciences, Madrid.",

year = "2024",

month = jan,

day = "1",

doi = "10.1007/s13398-023-01531-6",

language = "American English",

volume = "118",

journal = "Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales - Serie A: Matematicas",

issn = "1578-7303",

publisher = "Springer-Verlag Italia s.r.l.",

number = "1",

}

TY - JOUR

T1 - A simple Efimov space with sequentially-nice space of probability measures

AU - Banakh, T.

AU - Gabriyelyan, S.

PY - 2024/1/1

Y1 - 2024/1/1

N2 - 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.

AB - 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.

KW - Efimov space

KW - Gelfand–Phillips property

KW - Grothendieck property

KW - Nonatomic probability measure

KW - Selective sequential pseudocompactness

KW - Space of probability measures

UR - http://www.scopus.com/inward/record.url?scp=85178246404&partnerID=8YFLogxK

U2 - 10.1007/s13398-023-01531-6

DO - 10.1007/s13398-023-01531-6

M3 - Article

SN - 1578-7303

VL - 118

JO - Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales - Serie A: Matematicas

JF - Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales - Serie A: Matematicas

IS - 1

M1 - 32

ER -

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

Abstract

Keywords

All Science Journal Classification (ASJC) codes

Access to Document

Other files and links

Fingerprint

Cite this