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Let $T$ be the shift transformation. Let $P$ be invariant for $T$ and also define a discrete state space Markov Chain. Let $C_{1},\ldots,C_{n}$ be the connected components of the Markov Chain. It follows from Dobrushin's Decomposition Theorem that there exist $(P_{i})_{i \leq n}$ ergodic measures that define discrete state Markov Chains in each of the $C_{i}$'s and a measure $\mu$ such that:

$P = \sum_{i}{P_{i}\mu(i)}$

Are there known generalizations of this decomposition when $P$ is an invariant measure and $P$ defines a continuous state space Markov Chain? That is, do there exist ergodic $P_{\lambda}$ which define continuous state Markov Chains and a measure $\mu$ such that:

$P = \int{P_{\lambda} \mu(d\lambda)}$

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    I don't understand yet. I think there exist ergodic measures for the shift operator which are not Markovian. Using the Ergodic Decomposition Theorem, I don't see how to guarantee that the $P_{i}$ are Markovian (as required in the statement).2012-12-06

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