also called metric or measure-theoretical entropy

Kolmogorov-Sinai entropy

See the related Topological entropy

For a Measure-theoretical dynamical system, the metric entropy of the system with respect to a partition $\alpha$ is defined to be the Entropy rate of the stochastic process resulting from the partition.

The metric entropy (aka (Kolmogorov–Sinai or measure-theoretical) is then the supremum of {the metric entropy with respect to $\alpha$} over all finite partitions $\alpha$.

Metric entropy provides the maximum average information per unit of time obtainable per unit of time, from the dynamical system.

See Amigo's book for details. He also gives a good example with the tent map.

Note his notation $\vee$ refers to the join of two sigma-algebras. See here