Join (sigma algebra)

In mathematics, a join or refinement of two sigma algebras is the coarsest sigma algebra containing both.[1]

Given two sigma sub-algebras $\mathcal{A}\subset\mathcal{C}$ and $\mathcal{B}\subset\mathcal{C}$, their join $\mathcal{A} \vee \mathcal{B}\subset\mathcal{C}$ is given by

$\mathcal{A} \vee \mathcal{B} = \{A_i \cap B_j \; | \; A_i\in\mathcal{A}, B_j\in\mathcal{B} \}$

The above defines the join on sub-sigma algebras of a single common sigma algebra $\mathcal{C}$, as it is not generally possible to define the join of two unrelated sigma algebras.

Likewise, given two partitions Q = { Q1, ..., Qk } and R = { R1, ..., Rm } of a common set S, their join or refinement $\scriptstyle Q \vee R$ is given as

$Q \vee R = \{Q_i \cap R_j\ |\ i=1,\ldots,k,\ j=1,\ldots,m\}.\,$

The two definitions are equivalent: The unions of elements from a partition of a set generate a sigma algebra. Likewise, intersections of elements from a sigma algebra result in a partition of a set.