Kolmogorov automorphism

In mathematics, a Kolmogorov automorphism, K"-automorphism, K"-shift or K"-system is an invertible, measure-preserving automorphism defined on a standard probability space that obeys Kolmogorov's zero–one law. All Bernoulli automorphisms are "K"-automorphisms (one says they have the K"-property), but not vice versa. Many ergodic dynamical systems have been shown to have the "K"-property, although more recent research has shown that many of these are in fact Bernoulli automorphisms.
Although the definition of the "K"-property seems reasonably general, it stands in sharp distinction to the Bernoulli automorphism. In particular, the Ornstein isomorphism theorem does not apply to "K"-systems, and so the entropy is not sufficient to classify such systems – there exist uncountably many non-isomorphic "K"-systems with the same entropy. In essence, the collection of "K"-systems is large, messy and uncategorized; whereas the "B"-automorphisms are 'completely' described by Ornstein theory.
Formal definition.
Let formula_1 be a standard probability space, and let formula_2 be an invertible, measure-preserving transformation. Then formula_2 is called a "K"-automorphism, "K"-transform or "K"-shift, if there exists a sub-sigma algebra formula_4 such that the following three properties hold:
Here, the symbol formula_8 is the join of sigma algebras, while formula_9 is set intersection. The equality should be understood as holding almost everywhere, that is, differing at most on a set of measure zero.
Properties.
Assuming that the sigma algebra is not trivial, that is, if formula_10, then formula_11 It follows that "K"-automorphisms are strong mixing.
All Bernoulli automorphisms are "K"-automorphisms, but not "vice versa".