Exchangeable measures for subshifts

Let $\text{Om}$ be a Borel subset of $S^{\mathbb{N}}$ where $S$ is countable. A measure is called exchangeable on $\text{Om}$, if it is supported on $\text{Om}$ and is invariant under every Borel automorphism of $\text{Om}$ which permutes at most finitely many coordinates. De-Finetti’s theorem characterizes these measures when $\Om=S^\Bbb N$\mathrm{.}Weapplytheergodictheoryofequivalencerelationstostudythecase$\Om\neq S^\Bbb N$,andobtainversionsofthistheoremwhen$\Om$ is a countable state Markov shift, and when $\text{Om}$ is the collection of beta expansions of real numbers in $[0,1]$ (a non-Markovian constraint).