Amenable Pairs of Groups and Ergodic Actions and the Associated von Neumann Algebras

If X and Y are ergodic G-spaces, where G is a locally compact group, and X is an extension of Y, we study a notion of amenability for the pair $(X,Y)$. This simultaneously generalizes and expands upon previous work of the author concerning the notion of amenability in ergodic theory based upon fixed point properties of affine cocycles, and the work of Eymard on the conditional fixed point property for groups. We study the relations between this concept of amenability, properties of the von Neumann algebras associated to the actions by the Murray-von Neumann construction, and the existence of relatively invariant measures and conditional invariant means.