X $. Suppose that $ \fber \subseteq \cber $. There is a sequence of disjoint Borel sets $ A_{n}
  \subseteq X $ such that

1.  $ X = \bigsqcup_{n} A_{n} $;

2.  each $ A_{n} $ is $ \fber $-invariant;

3.  $ [x]_{\cber} \cap A_{n} = [x]_{\fber} $ for all $ x \in A_{n} $.

If there is a bound on $ [\fber:\cber] $ — the number of $ \fber $-classes in a $ \cber
  $-class — then the sequence $ (A_{n})_{n} $ can be taken to be finite.

Since $ \cber $ is smooth, it admits a Borel transversal $ B_{0} \subseteq X $. Set $ A_{0}
  = [B_{0}]_{\fber} $, and set recursively $ A_{n} = [B_{n}]_{\fber} $, where $ B_{n} $ is a Borel transversal for $ \cber $ restricted onto $ X \setminus \bigcup_{k< n} A_{k} $.

Back and forth construction {#sec:back-forth-constr}
===========================

For the proof of the following theorem it is convenient to introduce the notion of a tree of partitions. Let $ \mathbb{R}^{d} {\curvearrowright}X $ be a free flow on $ X $. A *tree of partitions* for the flow is a family of invariant Borel sets $ (\Omega_{s})_{s \in \mathbb{N}^{<\mathbb{N}}} $, $ \Omega_{s} \subseteq 