metric entropy
Let $(X,\mathcal{B},\mu )$ be a probability space, and $T:X\to X$ a measurepreserving transformation^{}. The entropy^{} of $T$ with respect to a finite measurable partition $\mathcal{P}$ is
$${h}_{\mu}(T,\mathcal{P})=\underset{n\to \mathrm{\infty}}{lim}{H}_{\mu}\left(\underset{k=0}{\overset{n1}{\bigvee}}{T}^{k}\mathcal{P}\right),$$ 
where ${H}_{\mu}$ is the entropy of a partition and $\vee $ denotes the join of partitions. The above limit always exists, although it can be $+\mathrm{\infty}$. The entropy of $T$ is then defined as
$${h}_{\mu}(T)=\underset{\mathcal{P}}{sup}{h}_{\mu}(T,\mathcal{P}),$$ 
with the supremum taken over all finite measurable partitions. Sometimes ${h}_{\mu}(T)$ is called the metric or measure theoretic entropy of $T$, to differentiate it from topological entropy.
Remarks.

1.
There is a natural correspondence between finite measurable partitions and finite sub$\sigma $algebras of $\mathcal{B}$. Each finite sub$\sigma $algebra is generated by a unique partition, and clearly each finite partition generates a finite $\sigma $algebra. Because of this, sometimes ${h}_{\mu}(T,\mathcal{P})$ is called the entropy of $T$ with respect to the $\sigma $algebra $\mathcal{P}$ generated by $\mathcal{P}$, and denoted by ${h}_{\mu}(T,\mathcal{P})$. This simplifies the notation in some instances.
Title  metric entropy 

Canonical name  MetricEntropy 
Date of creation  20130322 14:31:59 
Last modified on  20130322 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 