# 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

## Proof

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

Then:

 $\ds \map H {\AA \vee \CC}$ $=$ $\ds \map H {\AA \vee \CC \mid \NN}$ Conditional Entropy Given Trivial Sigma-Algebra is Entropy $\ds$ $=$ $\ds \map H {\AA \mid \NN} + \map H {\CC \mid \NN \vee \AA}$ Conditional Entropy of Join as Sum $\ds$ $=$ $\ds \map H \AA + \map H {\CC \mid \NN \vee \AA}$ Conditional Entropy Given Trivial Sigma-Algebra is Entropy $\ds$ $=$ $\ds \map H \AA + \map H {\CC \mid \AA}$ $\NN \vee \AA = \AA$ by Definition of Join of Finite Sub-Sigma-Algebras

$\blacksquare$