Conditional Entropy Given Trivial Sigma-Algebra is Entropy

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

Let $\AA \subseteq \Sigma$ be a finite sub-$\sigma$-algebra.

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

Then:
 * $\ds \map H {\AA \mid \NN} = \map H \AA$

where:
 * $\map H {\cdot \mid \cdot}$ denotes the conditional entropy
 * $\map H {\, \cdot \,}$ denotes the entropy