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