Behaviour of the reference measure on $\sf RCD$ spaces under charts

TL;DR

This paper demonstrates that on finite dimensional RCD spaces, the reference measure becomes absolutely continuous with respect to Lebesgue measure under rectifying charts, impacting the understanding of tangent space structures.

Contribution

It establishes the absolute continuity of the reference measure under charts on RCD spaces, advancing the geometric measure theory in this context.

Findings

Reference measure is absolutely continuous under charts

Implications for tangent space structure in RCD spaces

Utilizes recent measure structure results by De Philippis-Rindler

Abstract

Mondino and Naber recently proved that finite dimensional $\sf RCD$ spaces are rectifiable. Here we show that the push-forward of the reference measure under the charts built by them is absolutely continuous with respect to the Lebesgue measure. This result, read in conjunction with another recent work of us, has relevant implications on the structure of tangent spaces to $\sf RCD$ spaces. A key tool that we use is a recent paper by De Philippis-Rindler about the structure of measures on the Euclidean space.