Calculus of Variations and Geometric Measure Theory

M. Bagnara - L. Gennaioli - G. M. Leccese - E. Luongo

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

created by gennaioli on 03 Apr 2022
modified on 23 Mar 2023


Published Paper

Inserted: 3 apr 2022
Last Updated: 23 mar 2023

Journal: Real Analysis Exchange
Year: 2022

ArXiv: 2203.03393 PDF


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.