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 $f\colon K\to X$ we have $\mathcal{ H}^{ h} (K\cap f(K))=0$. This is a measure theoretic analogue of a result of M. Elekes.
Categories: math.CA