metric entropy

Let (X,,μ) be a probability space, and T:XX a measure-preserving transformationPlanetmathPlanetmath. The entropyPlanetmathPlanetmath of T with respect to a finite measurable partition 𝒫 is


where Hμ is the entropy of a partition and denotes the join of partitions. The above limit always exists, although it can be +. The entropy of T is then defined as


with the supremum taken over all finite measurable partitions. Sometimes hμ(T) is called the metric or measure theoretic entropy of T, to differentiate it from topological entropy.


  1. 1.

    There is a natural correspondence between finite measurable partitions and finite sub-σ-algebras of . Each finite sub-σ-algebra is generated by a unique partition, and clearly each finite partition generates a finite σ-algebra. Because of this, sometimes hμ(T,𝒫) is called the entropy of T with respect to the σ-algebra 𝒫 generated by 𝒫, and denoted by hμ(T,𝒫). This simplifies the notation in some instances.

Title metric entropy
Canonical name MetricEntropy
Date of creation 2013-03-22 14:31:59
Last modified on 2013-03-22 14:31:59
Owner Koro (127)
Last modified by Koro (127)
Numerical id 6
Author Koro (127)
Entry type Definition
Classification msc 28D20
Classification msc 37A35
Synonym entropy
Synonym measure theoretic entropy