Samplings and observables. Invariants of metric measure spaces

TL;DR

This paper explores the extension of invariants of metric measure spaces to their Gromov sampling compactification using graph limit theories and introduces ultralimits as a key technical tool.

Contribution

It identifies elements of the sampling compactification with geometric objects and extends invariants to this space, introducing ultralimits for the first time in this context.

Findings

Elements of the compactification are characterized as geometric objects.

Invariants of metric measure spaces are extended to the compactification.

Ultralimits are introduced as a main technical tool.

Abstract

In the paper we are dealing with metric measure spaces of diameter at most one and of total measure one. Gromov introduced the sampling compactification of the set of these spaces. He asked whether the metric measure space invariants extend to the compactification. Using ideas of the newly developed theory of graph limits we identify the elements of the compactification with certain geometric objects and show how to extend various invariants to this space. We will introduce the notion of ultralimit of metric measure spaces, that will be the main technical tool of our paper.