Definition:Entropy of Measure-Preserving Transformation with respect to Finite Sub-Sigma Algebra

Definition

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

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

Let $\AA \subseteq \BB$ be a finite sub-$\sigma$-algebra.

Then the entropy of $T$ with respect to $\AA$ is defined as:

$\ds \map h {T, \AA} := \lim_{n \mathop \to \infty} \dfrac 1 n \map H {\bigvee_{k \mathop = 0}^{n - 1} T^{-k} \AA}$

where:

$\map H \cdot$ denotes the entropy

$\vee$ denotes the join

$T^{-n} \AA$ is the pullback finite $\sigma$-algebra of $\AA$ by $T^n$

Also see

• Entropy of Measure-Preserving Transformation with respect to Finite Sub-Sigma Algebra is Well-Defined: The above limit exists in $\R_{\ge 0}$.

Sources

• 2013: Peter Walters: An Introduction to Ergodic Theory (4th ed.) $4.4$: Entropy of Measure-Preserving Transformation