Generating Finite Partition Preserves Order

Theorem

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

Let $\BB, \CC \subseteq \Sigma$ be finite sub-$\sigma$-algebras.

Then:

$\BB \subseteq \CC \iff \map \xi \BB \le \map \xi \CC$

where:

$\map \xi \cdot$ denotes the generated finite partition

$\le$ denotes the order by refinement of partition.

Proof

This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Sources

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