Generalized Hausdorff measure for generic compact sets

TL;DR

This paper shows that in a Polish space, the typical compact set is either finite or has a well-defined, finite Hausdorff measure with respect to some continuous gauge function, and that weak contractions on such sets have measure-zero intersections.

Contribution

It establishes a generic property of compact sets in Polish spaces regarding their Hausdorff measure and the behavior of weak contractions, answering an open question.

Findings

Generic compact sets are either finite or have finite, positive Hausdorff measure with a continuous gauge.

Weak contractions on these sets have measure-zero intersections.

The results provide a measure-theoretic analogue of a known geometric theorem.

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