On weakly metrically transitive measures and nonmeasurable sets. (English) Zbl 1132.28001

Kharazishvili, A. B.; Kirtadze, A. P.

Real Anal. Exch. 32(2006-2007), No. 2, 553-562 (2007).

Let \(G\) be a group of transformations of a nonempty set \(E\). A \(G\)-invariant measure on \(E\) is said to be weakly metrically transitive with respect to \(G\), if for any \(\varepsilon>0\) there exists a \(\mu\)-measurable set \(X\) with \(\mu(X)<\varepsilon\) and a countable subgroup \(H\) of \(G\) such that \(\mu(E\setminus\bigcup\{h(X):h\in H\})=0\). The author shows that some analog of Minkowski’s method from geometric number theory enables to establish the existence of sets which are absolutely nonmeasurable with respect to weakly metrically transitive invariant measures.

Reviewer: Miroslav Repický (Košice)

MSC:

28A05	Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
28D05	Measure-preserving transformations

Keywords:

invariant measure; convex set; weakly metrically transitive measure; extension of measure; \(G\)-selector