For a Borel action of a locally compact second countable group G on a standard Borel space S, are the orbits always Borel?

Question

In his book "Ergodic Theory and Semisimple Groups" Robert Zimmer opens Chapter 2 by discussing the situation of a locally compact second countable group $G$ with a Borel action on a standard Borel space $S$ and claims that the orbits are always Borel. For this he refers to Corollary 2.1.20 which establishes it in the case where $G$ is $\sigma$-compact, $S$ is a topological space, and $G$ acts continuously. What about with the original hypotheses? Is this result known?

Answer 1

It's definitely true. One way you could do it is that a locally compact second countable group $G$ is Polish (separable, completely metrizable), and if you have a Borel action of the Polish group $G$ on a standard Borel space $S$, there exists some Polish topology on $S$ so that the same action is continuous without changing the topology on $G$ (although that's a very tough theorem of Becker and Kechris). Then since that locally compact group is also $\sigma$-compact, you've got it.