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 $$h_\mu(T,\mathcal{P})=\lim_{n\to\infty}H_\mu\left(\bigvee_{k=0}^{n-1} T^{-k}\mathcal{P}\right),$$ 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 $\mathscr{B}$ 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.