Whole-space Axiom (Sigma Algebra)

probability-theory

Definition

Whole-space Axiom

Let $\mathfrak{S}\subseteq \mathcal{P}(\Omega )$. The whole-space axiom requires

$$\Omega \in \mathfrak{S}.$$

Thus the family contains the entire underlying space. It is one of the defining axioms of a sigma algebra.

Contrast

Membership of the whole space is the boundary

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

• Holds: ${\mathfrak{S}}_{+}=\{\{1\},\{2\},\Omega \}$ contains $\Omega$.
• Fails: ${\mathfrak{S}}_{-}=\{\{1\},\{2\}\}$ does not contain $\Omega$.

Although the members of ${\mathfrak{S}}_{-}$ collectively cover $\Omega$, their union is not itself a member. The whole-space axiom requires $\Omega \in \mathfrak{S}$ explicitly.