Locally Compact Groups and Haar Measure

Abstract

This chapter is intended to summarize some results needed later. We state an existence and uniqueness theorem for Haar measure but rather than copy a proof we leave this result unproven. See for example Halmos (1950) or Loomis (1953). The applications made in these notes are to matrix groups in their usual metric topology. Hence all topologies used in applications are Hausdorff topologies with countable base for open sets. More generality will be found in Loomis, op. cit., or Nachbin (1965). The manifolds discussed later are analytic manifolds for which the invariant measures can be given explicit representations using differential forms. Since the existence of invariant measures will usually be shown by explicit construction the part of the theory important to this book is usually the uniqueness part.