Measuring sets with translation invariant Borel measures

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Following Davies, Elekes and Keleti, we study measured sets, i.e. Borel sets B in R (or in a Polish group) for which there is a translation invariant Borel measure assigning positive and σ-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⊂ R of the second category is larger than any nullset A⊂ R in the sense that there are partitions B=B₁∪ B₂, A=A₁∪ A₂ and gauge functions g₁, g₂ such that the Hausdorff measures satisfy Hᵍⁱ(Bᵢ)=1 and Hᵍⁱ(Aᵢ)=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 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 σ-compact additive subgroup of 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-σ-finite for every Hausdorff measure of arbitrary gauge function.