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 $T$