Definition:Finite Partition Generated by 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 finite partition generated by $\AA$ is defined as:
 * $\ds \map \xi \AA := \set {\bigcap_{i \mathop = 1}^n B_i : B_i \in \set {A_i, \Omega \setminus A_i} } \setminus \set \O$

where $\AA = \set {A_1, \ldots, A_n}$.