Definition:Conditional Entropy of Finite Sub-Sigma-Algebra

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

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

The (conditional) entropy of $\AA$ given $\BB$ is defined as:
 * $\ds \map H {\AA \vert \BB} := \map H {\map\xi\AA \vert \map\xi\BB}$

where:
 * $\map H {\cdot\vert\cdot}$ on the denotes the conditional entropy of finite partitions
 * $\map\xi\cdot$ denotes the generated finite partition