Conditional Entropy of Join as Sum/Corollary 1

Corollary to Conditional Entropy of Join as Sum

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

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

Then:

$\map H {\AA \vee \CC} = \map H {\AA} + \map H {\CC \mid \AA} $

where:

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

$\map H \cdot$ denotes the entropy

$\vee$ denotes the join

Let $\NN := \set {\O, \Omega}$ be the trivial $\sigma$-algebra.

Then:

$\ds \map H {\AA \vee \CC}$

$=$

$\ds \map H {\AA \vee \CC \mid \NN}$

$\ds $

$=$

$\ds \map H {\AA \mid \NN} + \map H {\CC \mid \NN \vee \AA}$

$\ds $

$=$

$\ds \map H \AA + \map H {\CC \mid \NN \vee \AA}$

$\ds $

$=$

$\ds \map H \AA + \map H {\CC \mid \AA}$

$\NN \vee \AA = \AA$ by Definition of Join of Finite Sub-Sigma-Algebras

$\blacksquare$