The observable diameter of metric measure spaces and the existence of points of positive measures

2024/06/27 by Shun Oshima, Oshima, Shun
Mathematics · #53C23 #Advanced Banach Space Theory #Advanced Topology and Set Theory #FOS: Mathematics #Mathematical Dynamics and Fractals #Metric Geometry (math.MG)

paper · pdf · doi:10.48550/arxiv.2406.18863

openalex publication_date 2024/06/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A metric measure space is a metric space with a Borel measure. In Gromov's theory of metric measure spaces, there are important invariants called the partial diameter and the observable diameter. We obtain the result that the partial diameter or the observable diameter equals zero if and only if there exists a point that has positive measure.

Related

• Samplings and observables. Invariants of metric measure spaces
• Gromov's Problem: Bound the Expansion Coefficient from below in terms of the Observable Diameter of a Metric Measure Space, and its Diameter Bounds
• Metric measure geometry
• Topological aspects of the space of metric measure spaces
• Borel regularity is equivalent to Lusin's theorem and the existence of Borel representatives