Ultralimits of pointed metric measure spaces

TL;DR

This paper investigates ultralimits of pointed metric measure spaces, establishing their existence, relation to Gromov-Hausdorff convergence, and introducing a new convergence notion with a compactness theorem.

Contribution

It introduces a weaker convergence variant, proves ultralimits exist under mild conditions, and links ultralimits with direct and inverse limits in the metric measure space category.

Findings

Ultralimits exist under mild assumptions.

A new convergence notion is shown to be equivalent to pointed measured Gromov-Hausdorff convergence.

Established a compactness theorem for the new convergence variant.

Abstract

The aim of this paper is to study ultralimits of pointed metric measure spaces (possibly unbounded and having infinite mass). We prove that ultralimits exist under mild assumptions and are consistent with the pointed measured Gromov-Hausdorff convergence. We also introduce a weaker variant of pointed measured Gromov-Hausdorff convergence, for which we can prove a version of Gromov's compactness theorem by using the ultralimit machinery. This compactness result shows that, a posteriori, our newly introduced notion of convergence is equivalent to the pointed measured Gromov one. Another byproduct of our ultralimit construction is the identification of direct and inverse limits in the category of pointed metric measure spaces.