Generalized Hausdorff measure for generic compact sets (Q2860899)

The authors show that in a Polish space \(X\) the generic compact subset \(K\) of \(X\) (in the sense of Baire category) is either finite or there exists a continuous gauge function \(h\) such that the \(h-\)Hausdorff measure of \(K\) is positive and finite. \textit{R. O. Davies} [Mathematika 18, 161--162 (1971; Zbl 0229.28013)] constructed a Cantor set \(K\subset\mathbb{R}\), for which there is no gauge function \(h\) such that the \(h-\)Hausdorff measure of \(K\) is positive and finite. Later on, \textit{C. Cabrelli, U. B. Darji} and \textit{U. M. Molter} [``Visible and invisible Cantor sets'', in: Excursions in Harmonic Analysis, Volume 2: The February Fourier Talks at the Norbert Wiener Center, edited by Travis D. Andrews, Radu Balan, John J. Benedetto, Wojciech Czaja and Kasso A. Okoudjou, Springer, 11--22 (2013)] defined \(\mathcal{H}\)-visible sets and proved that the set of \(\mathcal{H}\)-visible compact sets is dense in the space of all non-empty compact subsets of \(\mathbb{R}\) endowed with the Hausdorff metric. Whether the generic compact set \(K\subset \mathbb{R}\) is \(\mathcal{H}\)-visible was left open at that point. The main theorem of this paper generalizes the results of Cabrelli, Darji, and Molter and gives an affirmative answer to the question mentioned previously. The authors also prove that for every weak contraction \(f:K\to X\) the set \(K\cap f(K)\) has zero \(h\)-Hausdorff measure. The latter is a somewhat stronger measure theoretic analogue of a result of [\textit{M. Elekes}, Proc. Am. Math. Soc. 137, No. 9, 3139--3146 (2009; Zbl 1173.54019)].