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
2.A note on Hausdorff measures of self-similar sets in Rd (arXiv)
Author : : Cai-Yun Ma, Yu-Feng Wu
Abstract : We prove that for all s∈(0,d) and c∈(0,1) there exists a self-similar set E⊂Rd with Hausdorff dimension s such that Hs(E)=c|E|s. This answers a question raised by Zhiying Wen