Generalized Hausdorff measure for generic compact sets

Abstract

Let $X$ be a Polish space. We prove that the generic compact set $K\subseteq X$ (in the sense of Baire category) is either finite or there is a continuous gauge function $h$ such that $0<{\mathcal{H}}^h(K)<\infty$, where ${\mathcal{H}}^h$ denotes the $h$-Hausdorff measure. This answers a question of C. Cabrelli, U. B. Darji, and U. M. Molter. Moreover, for every weak contraction $f:K\to X$ we have ${\mathcal{H}}^h(K\cap f(K))=0$. This is a measure theoretic analogue of a result of M. Elekes.

arXiv comment8 pages