Generating Finite Partition Preserves Order

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Theorem

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

Let $\BB, \CC \subseteq \Sigma$ be finite sub-$\sigma$-algebras.


Then:

$\BB \subseteq \CC \iff \map \xi \BB \le \map \xi \CC$

where:

$\map \xi \cdot$ denotes the generated finite partition
$\le$ denotes the order by refinement of partition.


Proof



Sources