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arXiv · 1807.02664

A note on topological dimension, Hausdorff measure, and rectifiability

Guy C. David·Enrico Le Donne
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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.

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Guy C. David, Enrico Le Donne. 2018-07-07. A note on topological dimension, Hausdorff measure, and rectifiability. https://arxiv.org/abs/1807.02664

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