Generating Finite Sub-Sigma-Algebra Preserves Order

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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.


Proof



Sources