On the Hausdorff Measure of $\R^n$ 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.