2015/04/10 by Máthé, András
#28A78 #28C10 (Primary) #46B15 (Secondary) #FOS: Mathematics #Functional Analysis (math.FA) #General Topology (math.GN)
paper · doi:10.48550/arxiv.1504.02765
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.