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

$\blacksquare$