Generated Finite Sub-Sigma-Algebra of Generated Finite Partition is Itself

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

Let $\BB\subseteq \Sigma$ be a finite sub-$\sigma$-algebra.

Then:
 * $\map \sigma {\map \xi \BB} = \BB$

where:
 * $\map\xi\cdot$ denotes the generated finite partition
 * $\map\sigma\cdot$ denotes the generated $\sigma$-algebra