On the Hausdorff Measure of Rn with the Euclidean Topology

In this paper, we answer a question raised by David H. Fremlin about the Hausdorff measure of R2 with respect to a distance inducing the Euclidean topology. In particular we prove that the Hausdorff n-dimensional measure of Rn 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.