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arXiv · 2203.03393

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

Marco Bagnara·Luca Gennaioli·Giacomo Maria Leccese·Eliseo Luongo
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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.

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Marco Bagnara, Luca Gennaioli, Giacomo Maria Leccese, Eliseo Luongo. 2022-03-02. On the Hausdorff Measure of $\R^n$ with the Euclidean Topology. https://arxiv.org/abs/2203.03393

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