Borel sets which are null or non- [formula omitted]-finite for every translation invariant measure

Abstract

We show that the set of Liouville numbers is either null or non- σ -finite with respect to every translation invariant Borel measure on R , in particular, with respect to every Hausdorff measure H g with gauge function g. This answers a question of R.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 R that are not F σ 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.