  We study ergodic decompositions of Dirichlet spaces under intertwining via
unitary order isomorphisms. We show that the ergodic decomposition of a
quasi-regular Dirichlet space is unique up to a unique isomorphism of the
indexing space. Furthermore, every unitary order isomorphism intertwining two
quasi-regular Dirichlet spaces is decomposable over their ergodic
decompositions up to conjugation via an isomorphism of the corresponding
indexing spaces.
