Closure under Complements (Sigma Algebra)

probability-theory

Definition

Closure under Complements

Let $\mathfrak{S}\subseteq \mathcal{P}(\Omega )$. It is closed under complements when

$$A\in \mathfrak{S}⟹A^c=\Omega \setminus A\in \mathfrak{S}.$$

Whenever the family contains an event, it therefore also contains the event that it does not occur. This is one of the defining axioms of a sigma algebra.

Contrast

Every member needs its complementary partner

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

• Holds: ${\mathfrak{S}}_{+}=\{\varnothing ,\Omega ,\{1\},\{2,3\}\}$ is closed under complements because its members occur in the pairs $\varnothing ,\Omega$ and $\{1\},\{2,3\}$.
• Fails: ${\mathfrak{S}}_{-}=\{\varnothing ,\Omega ,\{1\}\}$ is not closed under complements because $\{1\}\in {\mathfrak{S}}_{-}$ but $\{1{\}}^c=\{2,3\}\notin {\mathfrak{S}}_{-}$.

One missing complementary partner is enough for the axiom to fail.