Orbit structure and countable sections for actions of continuous groups

It is shown that if a second countable locally compact group G acts nonsingularly on an analytic measure space (S, μ), then there is a Borel subset E ? S such that EG is conull in S and each sG ∩ E is countable. It follows that the measure groupoid constructed from the equivalence relation s ～ sg on E may be simply described in terms of the measure groupoid made from the action of some countable group. Some simplifications are made in Mackey's theory of measure groupoids. A natural notion of “approximate finiteness” (AF) is introduced for nonsingular actions of G, and results are developed parallel to those for countable groups; several classes of examples arising naturally are shown to be AF. Results on “skew product” group actions are obtained, generalizing the countable case, and partially answering a question of Mackey. We also show that a group-measure space factor obtained from a continuous group action is isomorphic (as a von Neumann algebra) to one obtained from a discrete group action.