Closure under Countable Unions (Sigma Algebra)

probability-theory

Definition

Closure under Countable Unions

Let $\mathfrak{S}\subseteq \mathcal{P}(\Omega )$. It is closed under countable unions when every sequence $(A_i{)}_{i\in \mathbb{N}}$ of members of $\mathfrak{S}$ satisfies

$$(A_i{)}_{i\in \mathbb{N}}=\{A_1,A_2,A_3,\dots \}\subseteq \mathfrak{S}⟹\bigcup _{i=1}^{\infty }{A}_i\in \mathfrak{S}.$$

Thus combining countably many admitted events with “or” produces another admitted event. This is one of the defining axioms of a sigma algebra.

Contrast

Every countable union must remain in the family

Let $\Omega =\{1,2,3\}$. Compare

• Holds: ${\mathfrak{S}}_{+}=\{\varnothing ,\{1\},\{2\},\{1,2\},\Omega \}$ is closed under countable unions.
• Fails: ${\mathfrak{S}}_{-}=\{\varnothing ,\{1\},\{2\},\Omega \}$ is not closed under countable unions.

For the failing family, take $A_1=\{1\}$ and $A_i=\{2\}$ for every $i\ge 2$. Every $A_i$ belongs to ${\mathfrak{S}}_{-}$, but

$$\bigcup _{i=1}^{\infty }{A}_i=\{1,2\}\notin {\mathfrak{S}}_{-}.$$

A single countable sequence whose union leaves the family is enough for the axiom to fail.