Borel structure in groups and their duals

Abstract

Introduction. In the past decade or so much work has been done toward extending the classical theory of finite dimensional representations of compact groups a theory of (not necessarily finite dimensional) unitary representations of locally compact groups. Among the obstacles interfering with various aspects of this program is the lack of a suitable natural topology in the object; that is in the set of equivalence classes of irreducible representations. One can introduce natural topologies but none of them seem have reasonable properties except in extremely special cases. When the group is abelian for example the dual object itself is a locally compact abelian group. This paper is based on the observation that for certain purposes one can dispense with a topology in the dual object in favor of a and that there is a wide class of groups for which this weaker structure has very regular properties. If S is any topological space one defines a Borel (or Baire) subset of S be a member of the smallest family of subsets of S which includes the open sets and is closed with respect the formationi of complements and countable unions. The structure defined in S by its Bore! sets we may call the Borel structure of S. It is weaker than the topological structure in the sense that any one-to-one transformation of S onto S which preserves the topological structure also preserves the Borel structure whereas the converse is usually false. Of course a Borel structure may be defined without any reference a topology by simply singling out an arbitrary family of sets closed with respect the formation of complements and countable unions. Giving a set of mathematical objects a topology amounts giving it a sufficiently space-like structure so that one can speak of certain of the objects being near to or far away from certain others. Giving it a Borel structure amounts distinguishing a family of well behaved or definable subsets. In using the terms