A note on topological dimension, Hausdorff measure, and rectifiability

Abstract:

We give a sufficient condition for a general compact metric space to admit an $n$-rectifiable piece, as a consequence of a recent result of David Bate. 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 Csörnyei-Jones.