Measuring sets with translation invariant Borel measures

Following Davies, Elekes and Keleti, we study measured sets, i.e. Borel sets $B$ in $\mathbb{R}$ (or in a Polish group) for which there is a translation invariant Borel measure assigning positive and \sigma-finite measure to $B$. We investigate which sets can be written as a (disjoint) union of measured sets. We show that every Borel nullset $B\subset \mathbb{R}$ of the second category is larger than any nullset $A\subset \mathbb{R}$ in the sense that there are partitions $B=B_1\cup B_2$, $A=A_1\cup A_2$ and gauge functions $g_1, g_2$ such that the Hausdorff measures satisfy $H^{g_i}(B_i)=1$ and $H^{g_i}(A_i)=0$ ($i=1,2$). This implies that every Borel set of the second category is a union of two measured sets. We also present Borel and compact sets in $\mathbb{R}$ which are not a union of countably many measured sets. This is done in two steps. First we show that non-locally compact Polish groups are not a union of countably many measured sets. Then, to certain Banach spaces we associate a Borel and/or \sigma-compact additive subgroup of $\mathbb{R}$ which is not a union of countably many measured sets. It is also shown that there are measured sets which are null or non-\sigma-finite for every Hausdorff measure of arbitrary gauge function.