Relations of Finite Partition and Finite Sub-Sigma-Algebra

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.

Let:
 * $\map \xi \cdot$ denote the generated finite partition
 * $\map \sigma \cdot$ denote the generated $\sigma$-algebra
 * $\le$ denote the order by refinement of partition.

Then the following results hold: