Properties of Join

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

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

Then:
 * $\map \sigma {\eta \vee \gamma} = \map \sigma \eta \vee \map \sigma \gamma$

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