Generating Finite Sub-Sigma-Algebra Preserves Order

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

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

Then:
 * $\eta \le \gamma \iff \map \sigma \eta \subseteq \map \sigma \gamma$

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