F U N D a M E N T a Mathematicae 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 leftand right-invariant. Let G be a Polish abelian group. Christensen [C] calls a universally measurable set A ⊆ G Haar null if there exists a probability Borel measure μ on G such that μ(g + A) = 0 for all g ∈ G. It was proved in [C] that in case G is locally compact a universally measurable set is Haar null iff it is of Haar measure zero. Also, the union of a countable family of Haar null sets is Haar null, that is, Haar null sets constitute a σ-ideal. One of the first questions asked by Christensen in [C] was whether any family of mutually disjoint, universally measurable sets which are not Haar null is countable, as is the case when the group is Polish locally compact. This was answered in the negative by Dougherty [D] who constructed such uncountable families, for example, in all infinite dimensional Banach spaces. (Haar null sets are called “shy” in [D] following the terminology of [HSY].) This gives rise to the question whether the existence of such uncountable families characterizes non-locally-compact, Polish, abelian groups. We prove that this is indeed the case, that is, a Polish, abelian group is not locally compact iff 1991 Mathematics Subject Classification: 28C10, 43A05, 28A05.