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On Haar null sets

100%
Solecki S.
Fundamenta Mathematicae
|
1996
|
tom 149
|
nr 3
205-210
EN
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.
2
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Haar null and non-dominating sets

100%
Solecki S.
Fundamenta Mathematicae
|
2001
|
tom 170
|
nr 1-2
197-217
EN
We study the σ-ideal of Haar null sets on Polish groups. It is shown that on a non-locally compact Polish group with an invariant metric this σ-ideal is closely related, in a precise sense, to the σ-ideal of non-dominating subsets of $ω^{ω}$. Among other consequences, this result implies that the family of closed Haar null sets on a Polish group with an invariant metric is Borel in the Effros Borel structure if, and only if, the group is locally compact. This answers a question of Kechris. We also obtain results connecting Haar null sets on countable products of locally compact Polish groups with amenability of the factor groups.
3
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Filters and sequences

100%
Solecki S.
Fundamenta Mathematicae
|
2000
|
tom 163
|
nr 3
215-228
EN
We consider two situations which relate properties of filters with properties of the limit operators with respect to these filters. In the first one, we show that the space of sequences having limits with respect to a $Π^0_3$ filter is itself $Π^0_3$ and therefore, by a result of Dobrowolski and Marciszewski, such spaces are topologically indistinguishable. This answers a question of Dobrowolski and Marciszewski. In the second one, we characterize universally measurable filters which fulfill Fatou's lemma.
4
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Strong continuity of invariant probability charges

63%
Luschgy H., Solecki S.
Colloquium Mathematicum
|
2004
|
tom 101
|
nr 1
135-142
EN
Consider a semigroup action on a set. We derive conditions, in terms of the induced action of the semigroup on {0,1}-valued probability charges, which ensure that all invariant probability charges are strongly continuous.
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