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

TL;DR

This paper investigates the Hausdorff measure of Euclidean spaces under distances that induce the Euclidean topology, establishing that the measure of al{R}^n is always non-zero in this context, with counterexamples for other topologies.

Contribution

It proves that the Hausdorff measure of al{R}^n is never zero when using distances inducing the Euclidean topology, answering a question by Fremlin.

Findings

Hausdorff measure of al{R}^n is always non-zero with Euclidean topology

Counterexamples show the result does not hold for arbitrary topologies

The measure's behavior depends on the underlying topology and metric

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.