Join of Finite Sub-Sigma-Algebras Generates Join of Finite Partitions

Theorem
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $\BB, \CC \subseteq \Sigma$ be finite sub-$\sigma$-algebras.

Then:
 * $\map \xi {\BB \vee \CC} = \map \xi \BB \vee \map \xi \CC$

where:
 * $\map\xi\cdot$ denotes the generated finite partition
 * $\vee$ on the denotes the join of finite sub-$\sigma$-algebras
 * $\vee$ on the denotes the join of finite partitions