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

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 $+\infty$. The entropy of $T$ is then defined as

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. 1.

   There is a natural correspondence between finite measurable partitions and finite sub-$\sigma$-algebras of $\mathscr{B}$. 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 $\mathscr{P}$ generated by $\mathcal{P}$, and denoted by $h_{\mu}(T,\mathscr{P})$. This simplifies the notation in some instances.