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 right hand side denotes the entropy of finite partition

Also see

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

Sources

• 2013: Peter Walters: An Introduction to Ergodic Theory (4th ed.) $4.2$: Entropy of a Partition