Properties of Join

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

Let $\eta, \gamma$ be finite partitions of $\Omega$.

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

Then:
 * $(1)\quad$ $\map \sigma {\eta \vee \gamma} = \map \sigma \eta \vee \map \sigma \gamma$
 * $(2)\quad$ $\map \xi {\BB \vee \CC} = \map \xi \BB \vee \map \xi \CC$

where:
 * $\map\xi\cdot$ denotes the generated finite partition
 * $\map\sigma\cdot$ denotes the generated $\sigma$-algebra
 * $\vee$ denotes either join of partitions or join of $\sigma$-algebras.