Let 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 $$ h_\mu(T,\mathcal{P})=\lim_{n\to\infty}H_\mu\left(\bigvee_{k=0}^{n-1} T^{-k}\mathcal{P}\right) $$ 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 $+\infty$ . The entropy of $T$ is then defined as $$ h_\mu(T) = \sup_{\mathcal{P}} h_\mu(T,\mathcal{P}) $$ 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 . 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 generated by $\mathcal{P}$ , and denoted by . This simplifies the notation in some instances.