metric entropy

Let $(X,ℬ,\mu )$ be a probability space, and $T:X\to X$ a measure-preserving transformation.The entropy of $T$ with respect to a finite measurable partition $𝒫$ is

$$h_{\mu }(T,𝒫)=\underset{n\to \mathrm{\infty }}{lim}{H}_{\mu }\left(\underset{k=0}{\overset{n-1}{\bigvee }}{T}^{-k}𝒫\right),$$

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

$$h_{\mu }(T)=\underset{𝒫}{sup}{h}_{\mu }(T,𝒫),$$

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 finitesub-$\sigma$-algebras of $ℬ$. Each finite sub-$\sigma$-algebra isgenerated by a unique partition, and clearly each finite partition generates a finite $\sigma$-algebra.Because of this, sometimes $h_{\mu }(T,𝒫)$ is called the entropy of $T$ with respect tothe $\sigma$-algebra $𝒫$ generated by $𝒫$, and denoted by $h_{\mu }(T,𝒫)$.This simplifies the notation in some instances.