Measure-Preserving Transformation Preserves Conditional Entropy/Corollary

Corollary to Measure-Preserving Transformation Preserves Conditional Entropy
Let $\struct {X, \BB, \mu}$ be a probability space.

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

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

Then:
 * $\map H {T^{-1} \AA} = \map H \AA$

where:
 * $\map H \cdot$ denotes the conditional entropy
 * $T^{-1} \AA$ is the pullback finite $\sigma$-algebra of $\AA$ by $T$

Proof
Let $\NN := \set {\O, \Omega}$ be the trivial $\sigma$-algebra.

Since:
 * $T^{-1} \sqbrk \O = \O$

and:
 * $T^{-1} \sqbrk \Omega = \Omega$

it follows from :
 * $T^{-1} \NN = \NN$

Therefore: