Measure-Preserving Transformation Preserves Conditional Entropy

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

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

Let $\AA, \DD \subseteq \Sigma$ be finite sub-$\sigma$-algebras.

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

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

Proof
By, we have:
 * $\map \xi {T^{-1} \AA} = T^{-1} {\map \xi \AA}$

for each finite sub-$\sigma$-algebras $\AA \subseteq \Sigma$.

Thus it suffices to show that the entropy of finite partition satisfies:
 * $\map H {T^{-1} \xi \mid T^{-1} \eta} = \map H {\xi \mid \eta}$

for all $\xi, \eta$ be finite partitions of $\Omega$.

This follows immediately from the fact that:
 * $\map \mu {T^{-1} A \cap T^{-1} B} = \map \mu {A \cap B}$

for all $A, B \in \Omega$.