Measure-Preserving Transformation Preserves Conditional Entropy

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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 $T$


Corollary

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


Proof

By Definition of Finite Partition Generated by Finite Sub-Sigma-Algebra, 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 $\Omega$.

Now:

\(\ds \map H {T^{-1} \xi \mid T^{-1} \eta}\) \(=\) \(\ds \sum_{\substack {B \mathop \in \eta \\ \map \mu {T^{-1} B} \mathop > 0} } \map \mu {T^{-1} B} \sum_{A \mathop \in \xi} \map \phi {\dfrac {\map \mu {T^{-1} A \cap T^{-1} B} } {\map \mu {T^{-1} B} } }\) Definition of Conditional Entropy of Finite Partitions
\(\ds \) \(=\) \(\ds \sum_{\substack {B \mathop \in \eta \\ \map \mu B \mathop > 0} } \map \mu B \sum_{A \mathop \in \xi} \map \phi {\dfrac {\map \mu {A \cap B} } {\map \mu B } }\) since $T$ is $\mu$-preserving
\(\ds \) \(=\) \(\ds \map H {\xi \mid \eta}\) Definition of Conditional Entropy of Finite Partitions

$\blacksquare$