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