On the measurability of orbits in Borel actions

We replace measure with category in an argument of G. W. Mackey to characterize closed subgroups H of a totally nonmeager, 2nd countable topological group G in terms of the quotient Borel structure G/H. As a corollary, we obtain an improved version of a theorem of C. Ryll- Nardzewski on the Borel measurability of orbits in continuous actions by Polish groups. In (9), G. W. Mackey gave an argument using Haar measures to prove the following: Assume G is locally compact topological group. If H is a subgroup such that the space G/H, formed by giving the set of (left) H-cosets the quotient Borel structure, is countably separated, then H is closed in G. We will show that this result can be obtained by an analogous argument using the theory of Baire category. The category version has a wider application and shows that the above statement remains true under the weaker assumption that G is totally nonmeager. In particular, it holds whenever G is topologically complete (in the sense of Cech). As a corollary, we will prove that a well-known theorem of C. Ryll-Nardzewski on the Borel measurability of orbits in continuous actions by Polish (separable, completely metrizable) groups holds for Borel actions as well. We will also show that a recent result of R. Vaught on decompositions