Join of Finite Sub-Sigma-Algebras Generates Join of Finite Partitions

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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:

$\map \xi {\BB \vee \CC} = \map \xi \BB \vee \map \xi \CC$

where:

$\map\xi\cdot$ denotes the generated finite partition
$\vee$ on the left hand side denotes the join of finite sub-$\sigma$-algebras
$\vee$ on the right hand side denotes the join of finite partitions


Proof



Sources