## 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 | |

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