Editing Markov chain (section)

=== Measure-preserving dynamical system ===
If a Markov chain has a stationary distribution, then it can be converted to a [[measure-preserving dynamical system]]: Let the probability space be <math>\Omega = \Sigma^\N</math>, where <math>\Sigma</math> is the set of all states for the Markov chain. Let the sigma-algebra on the probability space be generated by the cylinder sets. Let the probability measure be generated by the stationary distribution, and the Markov chain transition. Let <math>T: \Omega \to \Omega</math> be the shift operator: <math>T(X_0, X_1, \dots) = (X_1, \dots) </math>. Similarly we can construct such a dynamical system with <math>\Omega = \Sigma^\Z</math> instead.<ref>{{Cite book |last=Kallenberg |first=Olav |title=Foundations of modern probability |date=2002 |publisher=Springer |isbn=978-0-387-95313-7 |edition=2. ed., [Nachdr.] |series=Probability and its applications |location=New York, NY Berlin Heidelberg |at=Proposition 8.6 (page 145)}}</ref>

Since ''irreducible'' Markov chains with finite state spaces have a unique stationary distribution, the above construction is unambiguous for irreducible Markov chains.

In [[ergodic theory]], a measure-preserving dynamical system is called ''ergodic'' if any measurable subset <math>S</math> such that <math>T^{-1}(S) = S</math> implies <math>S = \emptyset</math> or <math>\Omega</math> (up to a null set).

The terminology is inconsistent. Given a Markov chain with a stationary distribution that is strictly positive on all states, the Markov chain is ''irreducible'' if its corresponding measure-preserving dynamical system is ''ergodic''.<ref name=":2">{{Cite web |last=Shalizi |first=Cosma |author-link=Cosma Shalizi |date=1 Dec 2023 |title=Ergodic Theory |url=http://bactra.org/notebooks/ergodic-theory.html |access-date=2024-02-01 |website=bactra.org}}</ref>