Conditional Entropy of Join as Sum/Corollary 5

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} \le \map H \AA + \map H \CC $

where:
 * $\map H \cdot$ denotes the conditional entropy
 * $\vee$ denotes the join

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

Then: