Sigma-algebra

In mathematics, a σ-algebra (or sigma-algebra) over a set X is a nonempty collection Σ of subsets of X (including X itself) that is closed under complementation and countable unions of its members. It is a Boolean algebra, completed to include countably infinite operations. The pair (X, Σ) is also a field of sets, sometimes called a σ-field or a measurable space.

Thus, if X = {a, b, c, d}, one possible sigma algebra on X is Σ = { ∅, {a, b}, {c, d}, {a, b, c, d} }.

The main use of σ-algebras is in the definition of measures on X. The concept is important in mathematical analysis and probability theory.

Definition and properties

Formally, a subset Σ of the power set of a set X is a σ-algebra if it has the following properties:

1. Σ is not empty,
2. If E is in Σ then so is the complement (X \ E) of E,
3. The union of countably many sets in Σ is also in Σ.

In other words, a family Σ of subsets of X is a σ-algebra if:

1. Σ is not empty,
2. Σ is closed under complements,
3. Σ is closed under countable unions.

From these axioms, it follows that X and the empty set are in Σ, and that the σ-algebra is also closed under countable intersections (via De Morgan's laws).

A measure on X is a function which assigns a real number to subsets of X; this can be thought of as making precise a notion of "size" or "volume" for sets. One might like to assign such a size to every subset of X, but the axiom of choice implies that when the size under consideration is standard length for subsets of the real line, then there exist sets known as Vitali sets for which no size exists. For this reason, one considers instead a smaller collection of privileged subsets of X whose measure is defined; these sets constitute the σ-algebra.

Elements of the σ-algebra are called measurable sets. An ordered pair (X, Σ), where X is a set and Σ is a σ-algebra over X, is called a measurable space. A function between two measurable spaces is called measurable if the preimage of every measurable set is measurable. The collection of measurable spaces forms a category with the measurable functions as morphisms. Measures are defined as certain types of functions from a σ-algebra to [0, ∞].

σ-algebras are sometimes denoted using capital letters from a Fraktur typeface. Thus, may be used to denote (X, Σ). Another common convention is to use calligraphic capital letters in place of Σ, thus is often used in place of (X, Σ). This is handy to avoid situations where Σ might be confused for the summation operator.

Generated σ-algebra

If U is an arbitrary family of subsets of X then we can form a special σ-algebra containing U, called the σ-algebra generated by U. We denote it by σ(U) and define it as follows.

First note that there is a σ-algebra over X that contains U, namely the power set of X. Now let Φ denote the family of all σ-algebras over X that contain U (that is, a σ-algebra Σ over X is in Φ if and only if U is a subset of Σ.) Noting that the intersection of an arbitrary collection of σ-algebras is again a σ-algebra we define σ(U) to be the intersection of all σ-algebras in Φ (i.e. the intersection of all σ-algebras containing U.) Defined in this way σ(U) is the smallest σ-algebra over X that contains U.

For a simple example, consider the set X = {1, 2, 3}. Then the σ-algebra generated by the subset {1} is σ({1}) = {∅, {1}, {2,3}, X}. By an abuse of notation, when the collection of subsets contains only one member, call it A, one may write σ(A) instead of σ({A}).

Examples

Let X be any set, then the following are σ-algebras over X:

• The family consisting only of the empty set and X (the minimal or trivial σ-algebra over X).
• The full power set of X.
• The collection of subsets of X which are countable or whose complements are countable (which is distinct from the power set of X if and only if X is uncountable.). This is the σ-algebra generated by the singletons of X.
• If {Σ_λ} is a family of σ-algebras over X indexed by λ then the intersection of all Σ_λ is a σ-algebra over X.

Examples for generated algebras

An important example is the Borel algebra over any topological space: the σ-algebra generated by the open sets (or, equivalently, by the closed sets). Note that this σ-algebra is not, in general, the whole power set. For a non-trivial example, see the Vitali set.

On the Euclidean space Rⁿ, another σ-algebra is of importance: that of all Lebesgue measurable sets. This σ-algebra contains more sets than the Borel σ-algebra on Rⁿ and is preferred in integration theory, as it gives a complete measure space.

See also

• Measurable function
• Sample space
• Separable sigma algebra
• Sigma ring
• Sigma additivity

External links

• Sigma Algebra from PlanetMath.