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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.

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