Discretely approximable locally compact groups and Jessen's theorem for nilmanifolds

A topological group G   is said to be approximable by discrete subgroups, if there exists a sequence of discrete subgroups (Hn)n∈N(Hn)n∈N of G such that, for any non-empty open set O of G, there exists an integer k   such that O∩Hn≠∅O∩Hn≠∅, for every n≥kn≥k. In this paper we shall prove that the identity component of a locally compact group approximable by discrete subgroups is nilpotent. For connected nilpotent Lie groups, explicit approximating sequences of discrete subgroups are given. As an application, we extend Jessen's theorem on Riemann sums for torus to the case of nilmanifolds.