Stationary and invariant densities and disintegration kernels

Let be a locally compact topological group, acting measurably on some Borel spaces and , and consider some jointly stationary random measures on and on such that a.s. Then there exists a stationary random kernel from to such that a.s. This follows from the existence of an invariant kernel from to such that whenever . Also included are some related results on stationary integration, absolute continuity, and ergodic decomposition.