Abstract
In this paper, we introduce the concept of measure-theoretic r-entropy of a continuous map on a compact metric space, and get the results as follows: 1. Measure-theoretic entropy is the limit of measuretheoretic r-entropy and topological entropy is the limit of topological r-entropy (r → 0); 2. Topological r-entropy is more than or equal to the supremum of 4r-entropy in the sense of Feldman’s definition, where the measure varies among all the ergodic Borel probability measures.
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