Chapter 2: Group actions and relatively invariant integrals

Abstract

In this chapter, group actions are reviewed and are illustrated with examples of relevance for statistical applications. Relatively invariant integrals (measures) are defined and examples are given. An important result, due to Weil, gives necessary and sufficient conditions for the existence and uniqueness of relatively invariant integrals when the group action is transitive. A discussion of invariant and equivariant functions closes out the chapter.