Relations of Finite Partition and Finite Sub-Sigma-Algebra

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