On Haar null sets

Sławomir Solecki

Fundamenta Mathematicae (1996)

  • Volume: 149, Issue: 3, page 205-210
  • ISSN: 0016-2736

Abstract

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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.

How to cite

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Solecki, Sławomir. "On Haar null sets." Fundamenta Mathematicae 149.3 (1996): 205-210. <http://eudml.org/doc/212119>.

@article{Solecki1996,
abstract = {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.},
author = {Solecki, Sławomir},
journal = {Fundamenta Mathematicae},
keywords = {Haar null sets; countable chain condition; non-locally-compact groups; Polish groups},
language = {eng},
number = {3},
pages = {205-210},
title = {On Haar null sets},
url = {http://eudml.org/doc/212119},
volume = {149},
year = {1996},
}

TY - JOUR
AU - Solecki, Sławomir
TI - On Haar null sets
JO - Fundamenta Mathematicae
PY - 1996
VL - 149
IS - 3
SP - 205
EP - 210
AB - 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.
LA - eng
KW - Haar null sets; countable chain condition; non-locally-compact groups; Polish groups
UR - http://eudml.org/doc/212119
ER -

References

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  1. [B] M. Balcerzak, Can ideals without ccc be interesting? Topology Appl. 55 (1994), 251-260. Zbl0795.54052
  2. [C] J. P. R. Christensen, On sets of Haar measure zero in abelian Polish groups, Israel J. Math. 13 (1972), 255-260. 
  3. [De] C. Dellacherie, Capacities and analytic sets, in: Cabal Seminar 77-79, Lecture Notes in Math. 839, Springer, 1981, 1-31. 
  4. [D] R. Dougherty, Examples of non-shy sets, Fund. Math. 144 (1994), 73-88. Zbl0842.43006
  5. [HSY] B. R. Hunt, T. Sauer and J. A. Yorke, Prevalence: a translation-invariant "almost every" on infinite-dimensional spaces, Bull. Amer. Math. Soc. 27 (1992), 217-238. Zbl0763.28009
  6. [K] A. S. Kechris, Classical Descriptive Set Theory, Springer, 1995. 
  7. [TH-J]F. Topsœ and J. Hoffmann-Jørgensen, Analytic spaces and their applications, in: Analytic Sets, Academic Press, 1980, 317-401. 

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