A note on topological dimension, Hausdorff measure, and rectifiability

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.