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A note on topological dimension, Hausdorff measure, and rectifiability. (arXiv:1807.02664v1 [math.MG])
来源于:arXiv
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. 查看全文>>