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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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