Ergodic invariant measures for group-extensions fo dynamical systems

Let (X, x) be a measurable space. Let tau be a bi-measurable bijection from X onto X. Let phi be a measurable application from X to a second countable locally compact group G. We denote by tau(phi) the extension of tau, induced by phi, to the product space X x G. We describe the positive tau(phi)-invariant and ergodic measures on X x G. We also obtain a generalization of the cocycle reduction theorem of O. Sarig [5] to a general second countable locally group.