subsequence and are ordered: and likewise The -distance between and is where the supremum is being taken over all matches between and . This satisfies the triangle inequality only when and so is not quite a true metric; despite this, it is commonly called a "distance" in the literature. Generalizations Most of the properties of the Bernoulli scheme follow from the countable direct product, rather than from the finite base space. Thus, one may take the base space to be any standard probability space , and define the Bernoulli scheme as This works because the countable direct product of a standard probability space is again a standard probability space. As a further generalization, one may replace the integers by a countable discrete group , so that For this last case, the shift operator is replaced by the group action for group elements and understood as a function (any direct product can be understood to be the set of functions , as this is the exponential object). The measure is taken as the Haar measure, which is invariant under the group action: These generalizations are also commonly called Bernoulli schemes, as they still share most properties with the finite case. Properties Ya. Sinai demonstrated that the Kolmogorov entropy of a Bernoulli scheme is given by This may be seen as resulting from the general definition of the entropy of a Cartesian product of probability spaces, which follows from the asymptotic equipartition property. For the case of a general base space (i.e. a base space which is not countable), one typically considers the relative entropy. So, for example, if one has a countable partition of the base Y, such that , one may define the entropy as In general, this entropy will depend on the partition; however, for many dynamical systems, it is the case that the symbolic dynamics is independent of the partition (or rather, there are isomorphisms connecting the symbolic dynamics of different partitions, leaving the measure invariant), and so such systems can have a well-defined entropy independent of the partition. Ornstein Isomorphism Theorem The Ornstein isomorphism theorem states that two Bernoulli schemes with the same entropy are isomorphic. The result is sharp, in that very similar, non-scheme systems, such as Kolmogorov automorphisms, do not have this property. The Ornstein isomorphism theorem is in fact considerably deeper: it provides a simple criterion by which many different measure-preserving dynamical systems can