Properties of Join

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Theorem

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

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

Then:

$\map \sigma {\eta \vee \gamma} = \map \sigma \eta \vee \map \sigma \gamma$

where:

$\map \sigma \cdot$ denotes the generated $\sigma$-algebra

$\vee$ on the left hand side denotes the join of finite partitions

$\vee$ on the right hand side denotes the join of finite sub-$\sigma$-algebras

Proof

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Sources

• 2013: Peter Walters: An Introduction to Ergodic Theory (4th ed.) $4.1$: Partitions and Subalgebras

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