Definition:Join of Finite Partitions

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Definition

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

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

The join of $\xi$ and $\eta$ is the finite partition defined as:

$\ds \xi \vee \eta := \set {A \cap B : A \in \xi, B \in \eta}$

Similarly, given finite partitions $\xi_1, \ldots, \xi_n$ of $\Omega$:

$\ds \bigvee_{k \mathop = 1}^n \xi_k := \set { \bigcap_{k \mathop = 1}^n A_k \; : A_k \in \xi_k \; \text{for}\; \forall k }$

Also see

• Properties of Join
• Definition:Join of Open Covers

Sources

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