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Metric entropy and sufficient families. (English) Zbl 1032.28010
Summary: The aim of the present paper is to introduce the concept of sufficient families in the study of metric entropy $$h(\phi,{\mathcal N})$$ of an $$F$$-measure-preserving transformation $$\phi$$ relative to a sub $$\sigma$$-algebra $${\mathcal N}$$ of an $$F$$-dynamical system, $$\Phi= (X,{\mathcal M},m,\phi)$$ having finitely many atoms. Results including the Rokhlin inequality have been obtained. It is proved that if $${\mathcal N}$$ is a one-sided generator for $$[{\mathcal L}]$$ with respect to $$\phi$$ then the entropy $$h(\Phi,[{\mathcal L}])= 0$$. This result includes the corresponding classical result as a particular case.

##### MSC:
 28E10 Fuzzy measure theory 28D05 Measure-preserving transformations 37A35 Entropy and other invariants, isomorphism, classification in ergodic theory