Suppose we're given an action (possibly: ergodic) of a group G (possibly: uncountable) on a measure space $(X, \mu)$ (possibly: a standard probability space).
Question/reference request: is there a natural/standard way of using the von Neumann crossed product $L^{\infty} X \rtimes G$ to introduce a measure on the orbit space $X/G$ (or some representation thereof)?
Thoughts: the pushforward measure doesn't seem all that natural (e.g. the quotient Borel structure isn't always generated by the quotient topology, and for ergodic flows we get that each set of orbits is null or conull). I was wondering if there's some known alternative measure.