Haar nonmeasurable partitions of compact groups (Q1375788)

The paper deals with some aspects concerning the behaviour of compact groups. Mention must be made of the following results: Theorem 7. Let \(G\) be a nonmetrizable compact group. Then \(G\) can be partitioned into \(|G|\)-many pairwise disjoint dense homogeneously invariant \(\omega\)-bounded subsets each of which is of cardinality \(|G|\) and Haar nonmeasurable with full Haar outer measure. Theorem 8. Every infinite compact group \(G\) can be partitioned into \(|G|\) pairwise disjoint dense nonmeasurable sets of full Haar outer measure. At the end of the paper we find an important corollary, namely: Let \(A\) be an infinite discrete topological space. Then its Stone-Čech compactification \(\beta A\) can be partitioned into a collection of size \(|\beta A|\) pairwise disjoint \(\omega\)-bounded sets each of which is itself of size \(|\beta A|\).