Invariant Measurable Structures on Groups and Nonmeasurable Subgroups

Abstract

For any group $G$, the notion of an invariant measurable structure on $G$ is introduced. The following question is investigated: does there exist a subgroup of $G$ nonmeasurable with respect to this structure? It is demonstrated that, for an uncountable solvable group $G$, such a subgroup of $G$ always exists.