A note on topological dimension, Hausdorff measure, and rectifiability

TL;DR

This note establishes that compact metric spaces with finite Hausdorff measure and positive lower density contain rectifiable subsets, extending recent results and simplifying assumptions in geometric measure theory.

Contribution

It proves that such metric spaces necessarily contain rectifiable subsets of positive measure, generalizing previous results and removing the lower density assumption with recent advancements.

Findings

Spaces with finite Hausdorff measure have rectifiable subsets

Positive lower density implies existence of rectifiable parts

Recent results eliminate the need for lower density assumption

Abstract

The purpose of this note is to record a consequence, for general metric spaces, of a recent result of David Bate. We prove the following fact: Let $X$ be a compact metric space of topological dimension $n$. Suppose that the $n$-dimensional Hausdorff measure of $X$, $\mathcal H^n(X)$, is finite. Suppose further that the lower n-density of the measure $\mathcal H^n$ is positive, $\mathcal H^n$-almost everywhere in $X$. Then $X$ contains an $n$-rectifiable subset of positive $\mathcal H^n$-measure. Moreover, the assumption on the lower density is unnecessary if one uses recently announced results of Cs\"ornyei-Jones.