To make it work generally, we make use of the following classical ergodic decomposition theorem due to Varadarajan.

\[lem:ergodic-decomposition\] For any free Borel flow $ \mathbb{R}^{d} {\curvearrowright}X $ with $ \mathcal{E}(X) \ne {\varnothing}$ there exists a Borel surjection $ x \mapsto \mu_{x} $ from $ X $ onto $ \mathcal{E}(X) $ such that

(i) $ x \cber*[X] y \implies \mu_{x} = \mu_{y} $;

(ii) $ \nu \bigl(\bigl\{ x : \mu_{x} = \nu \bigr\}\bigr) = 1 $ for any $ \nu \in \mathcal{E}(X) $.

The following extension lemma will be used routinely through the back-and-forth construction. It is taken directly from [@MR1113569 Proposition 2.6].

\[lem:extension-lemma\] Let $ R, R' $ and $D_{i}, D_{i}' \subseteq \mathbb{R}^{d} $, $ 1 \le i \le n $, be smooth disks such that $ D_{i} \subseteq R $ and $ D_{i}' \subseteq R' $. Any family of orientation preserving diffeomorphisms $ f_{i} : D_{i} \to D_{i}' $ admits a common extension to an orientation preserving diffeomorphism $ f : R \to R' $.

Finally, we shall need the following easy fact from the theory of countable Borel equivalence relations.

\[lem:smooth-partition\] Let $ \cber, \fber $ be finite Borel equivalence relation on a standard Borel space $ 