Relations of Finite Partition and Finite Sub-Sigma-Algebra
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Theorem
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $\eta, \gamma$ be finite partitions of $\Omega$.
Let $\BB, \CC \subseteq \Sigma$ be finite sub-$\sigma$-algebras.
Let:
- $\map \xi \cdot$ denote the generated finite partition
- $\map \sigma \cdot$ denote the generated $\sigma$-algebra
- $\le$ denote the order by refinement of partition.
Then the following results hold:
Generated Finite Partition of Generated Finite Sub-Sigma-Algebra is Itself
- $\map \xi {\map \sigma \eta} = \eta$
Generated Finite Sub-Sigma-Algebra of Generated Finite Partition is Itself
- $\map \sigma {\map \xi \BB} = \BB$
Generating Finite Sub-Sigma-Algebra Preserves Order
- $\eta \le \gamma \iff \map \sigma \eta \subseteq \map \sigma \gamma$
Generating Finite Partition Preserves Order
- $\BB \subseteq \CC \iff \map \xi \BB \le \map \xi \CC$
Sources
- 2013: Peter Walters: An Introduction to Ergodic Theory (4th ed.) $4.1$: Partitions and Subalgebras