Extensions of ergodic group actions

In this paper we shall study extensions in the theory of ergodic actions of a locally compact group. If G is a locally compact group, by an ergodic G-space we mean a Lebesgue space (X,/) together with a Borel action of G on X, under which # is invariant and ergodic. If (X, #) and (Y, v) are ergodic G-spaces, (X,/) is called "an extension of (Y, v) and (Y, v) a factor of (X,/) if there is a Borel function p" X Y, commuting with the G-actions, such that p.(/) v. Various properties that one considers for a fixed ergodic G-space have as natural analogues properties of the triple (X, p, Y) in such a way as to reduce to the