The Steinhaus property in topological groups
by
Hans Weber
University of Udine, Italy
Coauthors: Enrico Zoli

Starting point of our considerations is the main result of P. Erdoes and S. Marcus, Sur la decomposition de l’espace Euclidean en ensembles homogènes, Acta Math. Acad. Sci. Hungar. 8, 443-452 (1957): Let E be a finitely dimensional Euclidean space; for any infinite cardinal number m not exceeding the cardinality of E there is a subgroup H of E of index m which is completely non-measurable and lacks the Baire property everywhere (i.e. for every measurable subset M of E of positive measure the intersection of H and M is non-measurable and for every non-empty open subset O of E the intersection of H and O doesn’t have the Baire property). Therefore the cosets of H form a decomposition of E in m “pathological” subsets. We give a new proof of the result of Erdoes and Marcus and generalize it replacing E by a locally compact abelian group G . Moreover, we examine finite pathological decompositions of G (inspired by pathological decompositions of the reals due to Sierpinski and Halmos). Our proofs are based on Steinhaus’ theorem and on a result on the existence of subgroups of an abelian group of fixed index. This algebraic result immediately yields as by-product a recent result of Comfort, Raczkowski and Trigos-Arrieta concerning the existence of “many” dense and non-measurable subgroups of an infinite compact abelian group. (They used such subgroups to produce precompact group topologies without non-trivial convergent sequences.) Another consequence of our approach is a result on the “thickness” of the complement of a union of translations of proper subgroups of a connected locally compact abelian group generalizing a result of K.Muthuvel (Application of covering sets, Colloq. Math. 80, 115-122 (1999) ) for the additive group of the reals.

Date received: May 13, 2005

Copyright © 2005 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts. Document # caqq-32.