Generated σ-algebra (by sets)

The σ-algebra generated by a family of sets or short generated σ-algebra or generated σ-field is a central concept in measure theory, a branch of mathematics that studies generalized notions of volumes. Generated σ-algebras are also common in probability theory, where they are for example used to model available information for stochastic processes via filtrations and provide the basis for the construction of the Borel σ-algebra.

Definition

Let [math]\displaystyle{ X }[/math] be some set. For any two families of sets [math]\displaystyle{ \mathcal C, \mathcal D }[/math] over [math]\displaystyle{ X }[/math], their intersection is defined as the family of sets

[math]\displaystyle{ \mathcal C \cap \mathcal D := \{ A \mid A \in \mathcal C \text{ and } A \in \mathcal D \} }[/math]

Then for any family of sets [math]\displaystyle{ \mathcal E }[/math] over [math]\displaystyle{ X }[/math], the σ-algebra generated by [math]\displaystyle{ \mathcal E }[/math] is defined as the smallest σ-algebra that contains [math]\displaystyle{ \mathcal E }[/math] and is noted as [math]\displaystyle{ \sigma(\mathcal E) }[/math]. The smallest σ-algebra is constructed by taking the intersections over all σ-algebras that contain [math]\displaystyle{ \mathcal E }[/math]:

[math]\displaystyle{ \sigma(\mathcal{E}) := \bigcap_{\mathcal{E} \subseteq \mathcal{A}\atop \mathcal{A} \,\,\sigma\text{-algebra}}\mathcal{A}. }[/math]

Comment

The definition of the generated σ-algebra is implicit, and most important σ-algebras that are constructed as generated σ-algebras cannot be written down explicitely. One such example is the Borel σ-algebra.

Nevertheless, the generated σ-algebra is well defined. This is since the intersection of any number of σ-algebras is not empty, because they all contain the set [math]\displaystyle{ X }[/math] and the empty set. The intersection of any number of σ-algebra is also a σ-algebra, see Sigma-algebra, so [math]\displaystyle{ \sigma(\mathcal E) }[/math] is again a σ-algebra.

Example

For some set [math]\displaystyle{ A \subset X }[/math], the generated σ-algebra is given by

[math]\displaystyle{ \sigma(A)=\{\emptyset, X, A , X\setminus A\}. }[/math]

This is because the empty set and the set [math]\displaystyle{ X }[/math] are contained in every σ-algebra. Since σ-algebras are closed under complements, the complement of [math]\displaystyle{ A }[/math] has to be included in the generated σ-algebra as well. This leads to the [math]\displaystyle{ \sigma(A) }[/math] as described above. This really is a σ-algebra since it satisfies all defining properties of a σ-algebra.

For a second example, assume the set [math]\displaystyle{ X }[/math] is at most countable and choose [math]\displaystyle{ \mathcal E }[/math] the set of all singletons: [math]\displaystyle{ \mathcal E:= \{ \{x \} \mid x\in X \} }[/math]

Then the generated σ-algebra is equal to the power set [math]\displaystyle{ \mathcal P(X) }[/math], since every subset [math]\displaystyle{ A }[/math] of [math]\displaystyle{ X }[/math] can be written as a countable union of singletons, so

[math]\displaystyle{ A \in \sigma(\mathcal E) \text{ for all } A \subset X, }[/math]

which leads to [math]\displaystyle{ \sigma(\mathcal E)= \mathcal P(X) }[/math].

Note that this would not be true for sets that are uncountable, since σ-algebras are just closed under countable set operations.