PARTITIONS OF LOCALLY COMPACT-GROUPS IN NONMEASURABLE SETS

Halmos [4] proves the existence of a partition of R into two Lebesgue non-measurable subsets, such that each of them is a translate of the other and their inner and outer measure differ everywhere as much as possible. In his short note [12], Thomas proves the same result, using a graph-theoretical argument. In this paper we compare Halmos' and Thomas' methods and we try to extend both of them to a nondiscrete, locally compact, T2, sigma-finite group, with respect to a left Haar measure. We obtain conditions, which assure that the group admits a partition into Haar nonmeasurable subsets, satisfying the above properties. In proving one of them, we use and extend a result by Kellerer [9].