below. A similar interpretation holds for continuous-time stochastic processes though the construction of the measurable structure of the action is more complicated. Ergodicity of Markov chains The dynamical system associated with a Markov chain Let be a finite set. A Markov chain on is defined by a matrix , where is the transition probability from to , so for every we have . A stationary measure for is a probability measure on such that ; that is for all . Using this data we can define a probability measure on the set with its product σ-algebra by giving the measures of the cylinders as follows: Stationarity of then means that the measure is invariant under the shift map . Criterion for ergodicity The measure is always ergodic for the shift map if the associated Markov chain is irreducible (any state can be reached with positive probability from any other state in a finite number of steps). The hypotheses above imply that there is a unique stationary measure for the Markov chain. In terms of the matrix a sufficient condition for this is that 1 be a simple eigenvalue of the matrix and all other eigenvalues of (in ) are of modulus <1. Note that in probability theory the Markov chain is called ergodic if in addition each state is aperiodic (the times where the return probability is positive are not multiples of a single integer >1). This is not necessary for the invariant measure to be ergodic; hence the notions of "ergodicity" for a Markov chain and the associated shift-invariant measure are different (the one for the chain is strictly stronger). Moreover the criterion is an "if and only if" if all communicating classes in the chain are recurrent and we consider all stationary measures. Examples Counting measure If for all then the stationary measure is the counting measure, the measure is the product of counting measures. The Markov chain is ergodic, so the shift example from above is a special case of the criterion. Non-ergodic Markov chains Markov chains with recurring communicating classes are not irreducible are not ergodic, and this can be seen immediately as follows. If are two distinct recurrent communicating classes there are nonzero stationary measures supported on respectively and the subsets and are both shift-invariant and of measure 1.2 for the invariant probability measure . A very simple example of that is the chain on