Sigma-Algebra Generated by Finite Partition is Finite Sub-Sigma-Algebra

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

Let $\xi$ be a finite partition of $\Omega$.

Let $\map \sigma \xi$ the generated $\sigma$-algebra by $\xi$.

Then, $\map \sigma \xi$ is a finite sub-$\sigma$-algebra of $\Sigma$.

Furthermore:
 * $\ds \map \sigma \xi = \set {\bigcup S: S \subseteq \xi}$