On 1-regular and 1-uniform metric measure spaces

TL;DR

This paper characterizes the geometric structure of 1-regular measures and classifies 1-uniform metric measure spaces, revealing their rectifiability properties and identifying exactly three such spaces.

Contribution

It provides a complete geometric characterization of 1-regular measures and classifies all 1-uniform metric measure spaces, including their rectifiability and unrectifiability.

Findings

Complete geometric characterization of rectifiable parts of 1-regular measures

Identification of exactly three 1-uniform metric measure spaces

Establishment of tangent space criteria for rectifiability

Abstract

A metric measure space $(X,\mu)$ is 1-regular if \[0< \lim_{r\to 0} \frac{\mu(B(x,r))}{r}<\infty\] for $\mu$-a.e $x\in X$. We give a complete geometric characterisation of the rectifiable and purely unrectifiable part of a 1-regular measure in terms of its tangent spaces. A special instance of a 1-regular metric measure space is a 1-uniform space $(Y,\nu)$, which satisfies $\nu(B(y,r))=r$ for all $y\in Y$ and $r>0$. We prove that there are exactly three 1-uniform metric measure spaces.