Definition:Kolmogorov-Sinai Entropy

Definition
Let $\struct {X, \BB, \mu}$ be a probability space.

Let $T: X \to X$ be a $\mu$-preserving transformation.

Then the entropy of $T$ is defined as:
 * $\ds \map h T := \sup \set {\map h {T, \AA} : \AA \; \text{finite sub-$\sigma$-algebra of} \; \BB }$

where:
 * $\ds \map h {T, \AA}$ denotes the entropy of $T$ with respect to $\AA$

Also known as
This is also called measure-theoretic entropy, Kolmogorov-Sinai entropy or KS entropy.

Remark

 * Entropy of Measure-Preserving Transformation may be Infinite: $\map h T$ may be $+\infty$.