An asymptotic definition of K-groups of automorphisms and a non-Bernoullian counter-example

  • Robert M. BurtonJr.
Article

Summary

Let G≅Zn be a group of measure preserving transformations of a Lebesgue space. J. P. Conze [1] has developed an entropy theory for such groups and described a class of groups obeying a form of the Kolmogorov “zero-one” law called K-groups. A Bernoulli group is a group isomorphic to the group of translates (shifts) of elements of the space \(\prod\limits_{g \in G} {X_g } \) with product measure where X g =X is a probability space. Bernoulli groups are also K-groups. Katznelson and Weiss [3] have shown entropy is a complete invariant for isomorphism classes of Bernoulli groups. We give an asymptotic definition of K-groups in terms of finite σ-algebras and justify this definition in terms of entropy and Conze's formulation. This definition s used to help us construct a K-group G≅Z n that is completely non-Bernoulli, that is one that contains no Bernoulli subgroup.

Keywords

Entropy Stochastic Process Probability Theory Mathematical Biology Probability Space 

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Bibliography

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Copyright information

© Springer-Verlag 1979

Authors and Affiliations

  • Robert M. BurtonJr.
    • 1
  1. 1.Department of MathematicsUniversity of ColoradoBoulderUSA

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