Measure and Topology in Groups

Abstract

In the preceding chapter we showed that in every locally compact group it is possible to introduce a left invariant Baire measure (or a left invariant, regular Borel measure) in an essentially unique manner. In this chapter we shall show that there are very close connections between the measure theoretic and the topological structures of such a group. In particular, in this section we shall establish some of the many results whose total effect is the assertion that not only is the measure determined by the topology, but that, conversely, all topological concepts may be described in measure theoretic terms. Throughout this section we shall assume that

X is a locally compact topological group, μ is a regular Haar measure on X, and ρ(E,F) = μ(E Δ F) for any two Borel sets E and F.