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Translation invariant ideals. (English) Zbl 1053.28006
The author proves a very general theorem on the existence of certain families of subsets of a group. This general theorem has some interesting consequences. It follows that the left Haar measure on any locally compact, second countable Abelian group $$G$$ admits a translation invariant extension which measures at least one selector of the family of cosets of any uncountable subgroup of $$G$$. This extends a result of A. Nowik [Real Anal. Exch. 22, No. 2, 814–817 (1996; Zbl 0943.28017)]. Another consequence is that, for a regular cardinal $$\kappa$$, any Abelian group $$G$$ carries a translation invariant ideal $${\mathfrak I}$$ with the property that ${\mathfrak I}^*:= \{X\subseteq G:\forall A\in{\mathfrak I}\;\exists g\in G\;(X+ g)\cap A=\emptyset\}= \{X\subseteq G:| X|< \kappa\}.$ This answers (for $$G=\mathbb{R}$$) a question of W. Seredynski [Colloq. Math. 57, No. 2, 203–219 (1989; Zbl 0694.28001)]. Moreover, it is proved – answering a question of Cichoń – that under MA there exits a subgroup of $$\mathbb{R}$$ of cardinality continuum whose all selectors are not Lebesgue measurable and do not have the property of Baire.
Reviewer: Hans Weber (Udine)

##### MSC:
 28C10 Set functions and measures on topological groups or semigroups, Haar measures, invariant measures 03E15 Descriptive set theory 28A05 Classes of sets (Borel fields, $$\sigma$$-rings, etc.), measurable sets, Suslin sets, analytic sets 43A05 Measures on groups and semigroups, etc. 03E50 Continuum hypothesis and Martin’s axiom
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##### References:
 [1] T. Bartoszyński and H. Judah,Set Theory. On the Structure of the Real Line, A K Peters, Wellesley, MA, 1995. [2] T. Carlson,Strong measure zero and strongly meager sets, Proceedings of the American Mathematical Society118 (1993), 577–586. · Zbl 0787.03037 · doi:10.1090/S0002-9939-1993-1139474-6 [3] J. Cichoń, A. Kharazishvili and B. Węglorz,On sets of Vitali’s type, Proceedings of the American Mathematical Society118 (1993), 1243–1250. · Zbl 0778.28001 [4] P. Erdös, K. Kunen and R.D. Mauldin,Some additive properties of sets of real numbers, Fundamenta Mathematicae113 (1981), 187–199. · Zbl 0482.28001 [5] P. Erdös and R.D. Mauldin,The nonexistence of certain invariant measures, Proceedings of the American Mathematical Society59 (1976), 321–322. · Zbl 0361.28013 [6] H. Friedman and M. Talagrand,Un ensemble singulier, Bulletin des Sciences Mathématiques104 (1980), 337–340. [7] F. Galvin, J. Mycielski and R. M. Solovay,Strong measure zero sets, Notices of the American Mathematical Society26 (1973), A-280. · Zbl 1417.03255 [8] A. B. Kharazishvili,On some types of invariant measures, Soviet Mathematics Doklady16 (1975), 681–684. · Zbl 0328.28011 [9] A. B. Kharazishvili,Transformation Groups and Invariant Measures. Set Theoretical Aspects, World Scientific, Singapore, 1998. · Zbl 1013.28012 [10] R. Laver,On the consistency of Borel’s conjecture, Acta Mathematica137 (1976), 151–169. · Zbl 0357.28003 · doi:10.1007/BF02392416 [11] A. Nowik,On a measure which measures at least one selector for every uncountable group, Real Analysis Exchange22 (1996/97), 814–817. · Zbl 0943.28017 [12] W. Seredyński,Some operations related to translations, Colloquium Mathematicum57 (1989), 203–219. [13] S. Shelah,Every null additive set is meager additive, Israel Journal of Mathematics89 (1995), 357–376. · Zbl 0819.03040 · doi:10.1007/BF02808209 [14] S. Solecki,Measurability properties of sets of Vitali’s type, Proceedings of the American Mathematical Society119 (1993), 897–902. · Zbl 0795.28010 [15] E. Szpilrajn (Marczewski),Sur l’extension de la mesure lebesguienne, Fundamenta Mathematicae25 (1935), 551–558. · Zbl 0013.00801
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