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.
[1] All Bernoulli automorphisms are K-automorphisms (one says they have the K-property), but not vice versa.
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.
is called a K-automorphism, K-transform or K-shift, if there exists a sub-sigma algebra
The equality should be understood as holding almost everywhere, that is, differing at most on a set of measure zero.
All Bernoulli automorphisms are K-automorphisms, but not vice versa.
Kolmogorov automorphisms are precisely the natural extensions of exact endomorphisms,[2] i.e. mappings