A measurable cross-section for orbits of a sample space under a free (exact) transformation group is shown to exist under topological regularity conditions. This is used to represent the sample space as essentially the product of a maximal invariant and an equivariant part, which implies Stein's representation for the density of the maximal invariant.
"Borel Cross-Sections and Maximal Invariants." Ann. Statist. 4 (5) 866 - 877, September, 1976. https://doi.org/10.1214/aos/1176343585