# 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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