Haar nonmeasurable partitions of compact groups

Abstract

We prove that a compact connected nonmetrizable group contains a proper dense $\omega$-bounded subgroup. This is then used to show that every compact nonmetrizable group $G$ can be partitioned into $|G|$-many pairwise disjoint dense topologically homogeneous $\omega$-bounded subsets each of cardinal $|G|$ and each Haar nonmeasurable with full Haar outermeasure. This allows us to then generalize an observation of Kakutani and Oxtoby and to conclude that each infinite compact group $G$ may be partitioned into a collection of $|G|$-many pairwise disjoint dense subsets of full Haar outermeasure. A corollary of these results is that the Stone-Cech compactification $\beta X$ of an infinite discrete space $X$ may be partitioned into $|\beta X|$ pairwise disjoint $\omega$-bounded subsets each of size $|\beta X|$.