Sigma Algebra

algebra probability-theory

Definition

Sigma Algebra

Let $\Omega$ be a set. A $\sigma$-algebra on $\Omega$ is a family of subsets

$$\mathfrak{S}\subseteq \mathcal{P}(\Omega )$$

satisfying the following axioms:

• Whole-space axiom: $\Omega \in \mathfrak{S}$.
• Closure under complements: $A\in \mathfrak{S}⟹A^c=\Omega \setminus A\in \mathfrak{S}$.
• Closure under countable unions: $(A_i{)}_{i\in \mathbb{N}}\subseteq \mathfrak{S}⟹{\bigcup }_{i=1}^{\infty }{A}_i\in \mathfrak{S}$.

These axioms also force closure under countable intersections. The pair $(\Omega ,\mathfrak{S})$ is a measurable space.

Axioms

Whole-space

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.

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Closure under complements

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.

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Closure under countable unions

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.

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Derived Closure Properties

The empty set belongs to every $\sigma$-algebra because

$$\varnothing ={\Omega }^c\in \mathfrak{S}.$$

For any sequence $(A_i{)}_{i\in \mathbb{N}}\subseteq \mathfrak{S}$, De Morgan’s law gives

$$\bigcap _{i=1}^{\infty }{A}_i={\left(\bigcup _{i=1}^{\infty }{A}_i^c\right)}^c\in \mathfrak{S}.$$

Hence a $\sigma$-algebra is closed under countable intersections and, in particular, under finite unions and intersections.

Extremal Sigma-Algebras

Smallest $\sigma$-algebra

The family

$$\{\varnothing ,\Omega \}$$

is the smallest $\sigma$-algebra on $\Omega$. It distinguishes only the impossible and certain events.

Largest $\sigma$-algebra

The power set

$$\mathcal{P}(\Omega )$$

is the largest $\sigma$-algebra on $\Omega$. It admits every subset of $\Omega$.