On Haar null sets
We prove that in Polish, abelian, non-locally-compact groups the family of Haar null sets of Christensen does not fulfil the countable chain condition, that is, there exists an uncountable family of pairwise disjoint universally measurable sets which are not Haar null. (Dougherty, answering an old question of Christensen, showed earlier that this was the case for some Polish, abelian, non-locally-compact groups.) Thus we obtain the following characterization of locally compact, abelian groups: Let G be a Polish, abelian group. Then the σ-ideal of Haar null sets satisfies the countable chain condition iff G is locally compact. We also show that in Polish, abelian, non-locally-compact groups analytic sets cannot be approximated up to Haar null sets by Borel, or even co-analytic, sets; however, each analytic Haar null set is contained in a Borel Haar null set. Actually, we prove all the above results for a class of groups which is much wider than the class of all Polish, abelian groups, namely for Polish groups admitting a metric which is both left- and right-invariant.
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