Groups and Group Actions

In group representation theory a group action by linear transformations is the fundamental notion. In this chapter, the authors provide some aspects related to this notion. A Polish group is a topological group whose topology is Polish. The Borel structure induced by a Polish group makes the Polish group a standard Borel group. A topological group is standard if the Borel structure induced by the topology is standard. The authors show that there are natural Borel structures on collections of representations. The existence of invariant measures on locally compact groups made possible the development of abstract representation theory for these groups. The theorem of Andre Weil shows these groups are in this sense the largest class possible. The authors consider direct integral decompositions for unitary representations of a second countable locally compact group G on a separable Hilbert space.