Suppose that $X$ is a standard Borel space (meaning it is endowed with a $\sigma$-algebra coming from some Polish topology on $X$) and $G$ is a Polish group acting in a Borel way on $X$. Denote by $E\_G$ the resulting orbit equivalence relation on $X$.


There are (at least) three competing definitions, that I know of, for this action or equivalence relation being *smooth*:


1. There is a standard Borel space $Y$ and a Borel measurable function $f:X\to Y$ such that for all $x,y\in X$, $xE\_G y$ if and only if $f(x)=f(y)$.
2. The quotient Borel structure on $X/E\_G$ is standard.
3. The quotient Borel structure on $X/E\_G$ is countably separated (meaning there is a sequence of Borel sets which separates points).


The first is what I'd call "the descriptive set theorists definition" (and can be found in any number of papers on Borel equivalence relations), while the third is "the ergodic theorists definition" (see, e.g., Zimmer, *Ergodic Theory and Semisimple Groups*). Clearly, the second implies the other two. And in the case when $G$ is countable, all three are equivalent (see Propositions 2.8-2.9 in <https://filippoc.people.uic.edu/2020/Spring/CBER.pdf>).


**My question:** Are these three definitions equivalent for general Polish groups $G$? If not, what about locally compact $G$? And is there a standard reference for these facts?



Competing definitions of smooth orbit equivalence relation
Let me turn my comment into an answer now that I have time to write some details. The short answer is that yes, those three definitions are equivalent, the long answer is below.


Let's forget about orbit equivalence relations and let's look at an arbitrary Borel equivalence relations $E$ on a Polish space $X$ for a second. Then we have (using a mix of the question's and Hjorth's terminology) the following three properties:


1. $E\leq\_B\mathrm{Id}\_{\Bbb R}$, that is there is a Borel $f\colon X\to\Bbb R$ such that for all $x,y\in X$, $xEy\iff f(x)=f(y)$. Here $\Bbb R$ can be replaced by any uncountable Polish space as they are all Borel isomorphic, or by a standard Borel space as in the question.
2. There exists a sequence $(A\_n)\_{n\in\Bbb N}$ of Borel $E$-invariant sets, such that for all $x,y\in X$, $xEy\iff\forall n(x\in A\_n\iff y\in A\_n)$, which is the same as the third condition in the question.
3. The quotient structure on $X/E$ generated by the Borel $E$-invariant sets is standard.


Then we have $(1)\iff (2)$ and $(3)\implies (2)$. If $E$ is the orbit equivalence relation induced by a continuous action of a Polish group, then all three conditions are equivalent. The first part is exercise 2.27 in Hjorth's *Classification and orbit equivalence relations*, let me give a sketch.


$(3)\implies (2)$ is obvious.  

$(1)\implies (2)$ Suppose $f$ is as in $(1)$ and let $(U\_n)\_{n\in\Bbb N}$ enumerate a countable basis for $\Bbb R$. Let $A\_n=f^{-1}(U\_n)$.  

$(2)\implies (1)$ Let $(A\_n)\_{n\in\Bbb N}$ be a sequence as in $(2)$. We reduce $E$ to $\mathrm{Id}\_{2^\Bbb N}$ with the map $\theta\colon X\to\Bbb 2^\Bbb N$, defined by $\theta(x)(n)=1\iff x\in A\_n$.


If $E=E^X\_G$ is the orbit equivalence relation induced by a continuous action $G\curvearrowright X$ of a Polish group $G$ we have that all three conditions are equivalent. This was proved by Burgess in *A selection theorem for group actions* (Pacific Journal of Mathematics, 80(2), pp.333-336.). To be precise Burgess shows that under the hypothesis above $E^X\_G$ admits a Borel transversal, that is a Borel set $A\subseteq X$ such that $|A\cap[x]\_{E^X\_G}|=1$ for all $x\in X$. Once such a transveral is available it is easy to check smoothness of $E^X\_G$ by reducing it to $\mathrm{Id}\_X$ with the function $f\colon X\to X$, $f(x)=$the unique element of $A$ which is $E^X\_G$-related to $x$.


