A note about charts built by Eriksson-Bique and Soultanis on metric measure spaces

TL;DR

This paper simplifies the proof of the existence of charts in metric measure spaces with finite Hausdorff dimension, aiding the understanding of measure-chart relations in RCD spaces.

Contribution

It provides an alternative, simpler proof of chart existence in metric measure spaces and clarifies measure-chart relations in RCD spaces.

Findings

Charts exist on sets of finite Hausdorff dimension

Simplified proof of chart construction

Insights into measure and chart relations in RCD spaces

Abstract

This note is motivated by recent studies by Eriksson-Bique and Soultanis about the construction of charts in general metric measure spaces. We analyze their construction and provide an alternative and simpler proof of the fact that these charts exist on sets of finite Hausdorff dimension. The observation made here offers also some simplification about the study of the relation between the reference measure and the charts in the setting of $\text{RCD}$ spaces.