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 $T$

Corollary

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

Proof

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