Borel sets which are null or non-$\sigma$-finite for every translation invariant measure
Classical Analysis and ODEs
2011-09-27 v1
Abstract
We show that the set of Liouville numbers is either null or non--finite with respect to every translation invariant Borel measure on , in particular, with respect to every Hausdorff measure with gauge function . This answers a question of D. Mauldin. We also show that some other simply defined Borel sets like non-normal or some Besicovitch-Eggleston numbers, as well as all Borel subgroups of that are not possess the above property. We prove that, apart from some trivial cases, the Borel class, Hausdorff or packing dimension of a Borel set with no such measure on it can be arbitrary.
Keywords
Cite
@article{arxiv.1109.5309,
title = {Borel sets which are null or non-$\sigma$-finite for every translation invariant measure},
author = {Márton Elekes and Tamás Keleti},
journal= {arXiv preprint arXiv:1109.5309},
year = {2011}
}