# Conditional Entropy of Join as Sum/Corollary 2

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## Corollary to Conditional Entropy of Join as Sum

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $\AA, \CC, \DD \subseteq \Sigma$ be finite sub-$\sigma$-algebras.

Then:

- $\AA \subseteq \CC \implies \map H {\AA \mid \DD} \le \map H {\CC \mid \DD} $

where:

- $\map H {\cdot \mid \cdot}$ denotes the conditional entropy
- $\vee$ denotes the join

## Proof

Let $\AA \subseteq \CC$.

Then:

\(\ds \map H {\CC \mid \DD}\) | \(=\) | \(\ds \map H {\AA \vee \CC \mid \DD}\) | as $\AA \vee \CC = \CC$ by Definition of Join of Finite Sub-Sigma-Algebras | |||||||||||

\(\ds \) | \(=\) | \(\ds \map H {\AA \mid \DD} + \map H {\CC \mid \DD \vee \AA}\) | Conditional Entropy of Join as Sum | |||||||||||

\(\ds \) | \(\ge\) | \(\ds \map H {\AA \mid \DD}\) | as $\map H { \CC \mid \DD \vee \AA } \ge 0$ by Definition of Conditional Entropy of Finite Sub-Sigma-Algebra |

$\blacksquare$

## Sources

- 2013: Peter Walters:
*An Introduction to Ergodic Theory*(4th ed.) $4.3$: Conditional Entropy