Universal measure zero, large Hausdorff dimension, and nearly Lipschitz maps

Volume 218 / 2012

Ondřej Zindulka Fundamenta Mathematicae 218 (2012), 95-119 MSC: 28A78, 54E35, 54F45. DOI: 10.4064/fm218-2-1


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.


  • Ondřej ZindulkaDepartment of Mathematics
    Faculty of Civil Engineering
    Czech Technical University
    Thákurova 7
    160 00 Praha 6, Czech Republic

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