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June 1997 Haar nonmeasurable partitions of compact groups
G.L. Itzkowitz, D. Shakhmatov
Tsukuba J. Math. 21(1): 251-262 (June 1997). DOI: 10.21099/tkbjm/1496163176

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 $|\beta X|$.

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G.L. Itzkowitz. D. Shakhmatov. "Haar nonmeasurable partitions of compact groups." Tsukuba J. Math. 21 (1) 251 - 262, June 1997. https://doi.org/10.21099/tkbjm/1496163176

Information

Published: June 1997
First available in Project Euclid: 30 May 2017

zbMATH: 0887.22004
MathSciNet: MR1467236
Digital Object Identifier: 10.21099/tkbjm/1496163176

Rights: Copyright © 1997 University of Tsukuba, Institute of Mathematics

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Vol.21 • No. 1 • June 1997
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