Borel sets which are null or non-σ-finite for every translation invariant measure

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 Hg 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 Fσ 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.