Definition:Finite Partition Generated by Finite Sub-Sigma-Algebra
Jump to navigation
Jump to search
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 $\AA = \set {A_1, \ldots, A_n}$.
Sources
- 2013: Peter Walters: An Introduction to Ergodic Theory (4th ed.) $4.1$: Partitions and Subalgebras