ABSTRACT
In this chapter, the authors define the notion of ergodicity for quasi-invariant measure. They need the more general notion of an ergodic measure which may not be quasi-invariant. The Borel structure generated by the quotient topology is countably separated. The quotient Borel structure is generated by any countable separating family of Borel sets. Hence the quotient topology generates the quotient Borel structure. That the topologies correspond can be shown using the fact that the topologies are defined in terms of kernels and hulls of representations and these are algebraic definitions. The authors note that the results on unitary representations of second countable locally compact groups and their direct integrals hold for nondegenerate representations of separable type I C* algebra on separable Hilbert spaces.
