Universal measure zero, large Hausdorff dimension, and nearly Lipschitz maps
Volume 218 / 2012
Fundamenta Mathematicae 218 (2012), 95-119
MSC: 28A78, 54E35, 54F45.
DOI: 10.4064/fm218-2-1
Abstract
We prove that each analytic set in $\mathbb{R}^n$ contains a universally null set of the same Hausdorff dimension and that each metric space contains a universally null set of Hausdorff dimension no less than the topological dimension of the space. Similar results also hold for universally meager sets.
An essential part of the construction involves an analysis of Lipschitz-like mappings of separable metric spaces onto Cantor cubes and self-similar sets.
