Definition:Join of Finite Sub-Sigma-Algebras
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Definition
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $\AA, \BB \subseteq \Sigma$ be finite sub-$\sigma$-algebras.
The join of $\AA$ and $\BB$ is the finite sub-$\sigma$-algebra defined as:
- $\ds \AA \vee \BB := \map \sigma {\AA \cup \BB}$
where $\map \sigma \cdot$ denotes the generated $\sigma$-algebra.
Also known as
The join of $\AA_0, \AA_1, \ldots, \AA_n \subseteq \Sigma$ is recursively defined:
- $\bigvee _{k=0}^n \AA_k := \begin{cases} \AA_0 & : n = 0 \\ \paren {\bigvee _{k=0}^{n-1} \AA_k} \vee \AA_n & : n > 0 \end{cases}$
Also see
Sources
- 2013: Peter Walters: An Introduction to Ergodic Theory (4th ed.) $4.1$: Partitions and Subalgebras