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

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

Also known as
The Kolmogorov-Sinai entropy is also known as:
 * metric entropy
 * measure-theoretic entropy
 * Kolmogorov entropy
 * KS entropy.

Also see

 * Kolmogorov-Sinai Entropy may be Infinite: $\map h T$ may be $+\infty$.