Definition:Refinement of Partition (Probability Theory)

Definition

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

Let $\xi, \eta$ be partitions of $\Omega$.

Then $\xi$ said to be a refinement of $\eta$ if and only if:

$\ds \forall A \in \eta : A = \bigcup \set {B \in \xi : B \subseteq A}$

It is written as:

$\eta \le \xi$

Sources

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