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