Bounded Scale Measure and Property A

TL;DR

This paper introduces bounded scale measure as a new coarse invariant, generalizes property A to large scale spaces with this measure, and establishes its invariance and properties.

Contribution

It defines bounded scale measure, proves its invariance, and extends property A to a broader class of large scale spaces with this measure.

Findings

Bounded scale measure is a coarse invariant unlike bounded geometry.

Equivalent definitions for spaces with bounded scale measure are provided.

Property A is generalized and shown to be a coarse invariant for these spaces.

Abstract

We introduce a generalization for bounded geometry that we call bounded scale measure. We show that bounded scale measure is a coarse invariant unlike bounded geometry. We then show equivalent definitions for spaces with bounded scale measure and show other properties that spaces with bounded scale measure satisfy. From there, we generalize property A from uniformly discrete metric spaces to large scale spaces with bounded geometry. Lastly, we construct a definition for property A for large scale spaces with bounded scale measure and show that this definition of property A is a coarse invariant.