Entropy of Measure-Preserving Transformation with respect to Finite Sub-Sigma Algebra is Well-Defined

Theorem

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 $\AA$:

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

Let:

$\ds a_n := \map H {\bigvee_{k \mathop = 0}^{n - 1} T^{-k} \AA}$

We need to show that the limit:

$\ds \lim_{n \mathop \to \infty} \frac {a_n} n$

exists.

In view of Fekete's Subadditive Lemma, it suffices to show the subadditivity of $\sequence {a_n}$.

To this end, let $m, n \ge 1$.

Then:

$\ds a_{m + n}$

$=$

$\ds \map H {\bigvee_{k \mathop = 0}^{m - 1} T^{-k} \AA \vee \bigvee_{k \mathop = m}^{m + n - 1} T^{-k} \AA }$

$\ds $

$\le$

$\ds \map H {\bigvee_{k \mathop = 0}^{m - 1} T^{-k} \AA} + \map H {\bigvee_{k \mathop = m}^{m + n - 1} T^{-k} \AA }$

$\ds $

$=$

$\ds \map H {\bigvee_{k \mathop = 0}^{m - 1} T^{-k} \AA} + \map H {\bigvee_{k \mathop = 0}^{n - 1} T^{-k} \AA }$

$\ds $

$=$

$\ds a_m + a_n$

$\blacksquare$