Definition:Entropy of Finite Sub-Sigma-Algebra

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

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

The entropy of $\AA$ is defined as:
 * $\ds \map H \AA := \map H {\map \xi \AA}$

where:
 * $\map \xi \AA$ is the finite partition generated by $\AA$
 * $\map H \cdot$ on the denotes the entropy of finite partition

Also see

 * Definition:Conditional Entropy of Finite Sub-Sigma-Algebra