Sigma Algebra

algebra probability-theory

Definition

Sigma Algebra

Let be a set. A -algebra on is a family of subsets

satisfying the following axioms:

• Whole-space axiom: .
• Closure under complements: .
• Closure under countable unions: .

These axioms also force closure under countable intersections. The pair is a measurable space.

Axioms

Whole-space

Definition

Whole-space Axiom

Let . The whole-space axiom requires

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 . It is closed under complements when

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 . It is closed under countable unions when every sequence of members of satisfies

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 -algebra because

For any sequence , De Morgan’s law gives

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

Extremal Sigma-Algebras

Smallest -algebra

The family

is the smallest -algebra on . It distinguishes only the impossible and certain events.

Largest -algebra

The power set

is the largest -algebra on . It admits every subset of .