Calculus of Variations and Geometric Measure Theory

G. David - E. Le Donne

A note on topological dimension, Hausdorff measure, and rectifiability

created by ledonne on 14 Sep 2018


Submitted Paper

Inserted: 14 sep 2018
Last Updated: 14 sep 2018

Year: 2018


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 $X$, $H^n(X)$, is finite. Suppose further that the lower $n$-density of the measure $H^n(X)$ is positive, $H^n(X)$-almost everywhere in $X$. Then $X$ contains an $n$-rectifiable subset of positive $H^n(X)$-measure. Moreover, the assumption on the lower density is unnecessary if one uses recently announced results of Csörnyei-Jones.

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