Translation invariant ideals (Q1424030)

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 \textit{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 \textit{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.