Definition:Join of Finite Sub-Sigma-Algebras

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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