  We show that the set of Liouville numbers is either null or
non-$\sigma$-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 $F_\sigma$ 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.

Classical Analysis and ODEs