# 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 Definition of Pullback Finite Sigma-Algebra:

$T^{-1} \NN = \NN$

Therefore:

 $\ds \map H \AA$ $=$ $\ds \map H {\AA \mid \NN}$ Conditional Entropy Given Trivial $\sigma$-Algebra is Entropy $\ds$ $=$ $\ds \map H {T^{-1} \AA \mid T^{-1} \NN}$ Measure-Preserving Transformation Preserves Conditional Entropy $\ds$ $=$ $\ds \map H {T^{-1} \AA \mid \NN}$ as $T^{-1} \NN = \NN$ $\ds$ $=$ $\ds \map H {T^{-1} \AA}$ Conditional Entropy Given Trivial $\sigma$-Algebra is Entropy

$\blacksquare$