On the Hausdorff Measure of ${\mathbb{R}}^n$ with the Euclidean Topology

Abstract

In this paper we answer a question raised by David H. Fremlin about the Hausdorff measure of ${\mathbb{R}}^2$ with respect to a distance inducing the Euclidean topology. In particular we prove that the Hausdorff $n$-dimensional measure of ${\mathbb{R}}^n$ is never $0$ when considering a distance inducing the Euclidean topology. Finally, we show via counterexamples that the previous result does not hold in general if we remove the assumption on the topology.

Corrected typos, added remarks, title changed