I am trying to solve exercise 1.9 in these notes, where we are asked to prove that $\mathcal{R} := \{ (x,g \cdot x) : x \in X, g \in G\}$ is a countable, probability measure preserving (p.m.p) equivalence relation whenever $G$ is a countable group acting on the standard probability space $(X, \mu)$ in a way that is p.m.p.
Clearly it is a countable equivalence relation, but I am having trouble verifying that $\mathcal{R}$ is a Borel subset of $X \times X$ (I tried showing that it is the preimage of a Borel set under a Borel map with no success) and that it satisfies the following property: any partial isomorphism $\theta : A \to B$ belonging to the full groupoid $[[\mathcal{R}]]$ of $\mathcal{R}$ is measure preserving (no ideas for this one).
EDIT: Okay. I think it's pretty easy to see why $\mathcal{R}$ is a Borel subset of $X \times X$. Indeed, given $g \in G$, $g$ is a Borel isomorphism of $X$ so, in particular, its graph $graph(g)$ is a Borel set. But it's pretty clear that $\mathcal{R} = \bigcup_{g \in G} graph(g)$ which is a countable union of Borel sets so it must be Borel itself, although I am not entirely sure yet why $graph(g)$ is also Borel...So, that knocks out the first part.