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.

Then:
 * $(1)\quad$ $\map \xi {\map \sigma \eta} = \eta$
 * $(2)\quad$ $\map \sigma {\map \xi \BB} = \BB$
 * $(3)\quad$ $\eta\le\gamma\iff\map\sigma\eta\subseteq\map\sigma\gamma$
 * $(4)\quad$ $\BB\subseteq\CC\iff\map\xi\BB\le\map\xi\CC$

where:
 * $\map\xi\cdot$ denotes the generated finite partition
 * $\map\sigma\cdot$ denotes the generated $\sigma$-algebra
 * $\le$ denotes the order by refinement of partition