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

$\blacksquare$


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