On the Hausdorff Measure of ℝ^n with the Euclidean Topology
- Authors: Marco Bagnara, Luca Gennaioli, Giacomo Maria Leccese, Eliseo Luongo
- Preprint year: 2022
- First public date: 2022-03-02
- arXiv: 2203.03393
- Status: Published
- Publication type: Journal article
- Publication year: 2023
- Journal: Real Analysis Exchange 48(1), 101–106 (2023)
- DOI: 10.14321/realanalexch.48.1.1649735306
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.