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 $\RR$, in particular, with respect to every Hausdorff measure $\iH^g$ with gauge function $g$. 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 $\RR$ 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.

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