Borel set

The Borel σ - algebra is a term from mathematics, which forms a hinge between the branches of topology and measure theory. Each topological space can be assigned to a σ - algebra in a unique way, which is called the associated Borel σ - algebra.

The term is named after the mathematician Émile Borel.

• 2.1 The Borel σ - algebra on separable metric spaces
• 2.2 The Borel σ - algebra on the real numbers
• 2.3 The Borel σ - algebra on finite dimensional real vector spaces

Definition

For a given topological space, the Borel σ - algebra is defined as the smallest σ - algebra containing the open sets of. The elements of this σ - algebra are called Borel sets.

Comments

• A σ - algebra on a base set is a set of subsets that contains the basic amount and the respect to complementation and countable union is complete. A basic amount together with a stated on their σ - algebra is also called the measuring room.
• The fact that there is exactly one such smallest σ -algebra is shown in the paragraph on the σ - operator. It is nothing but the intersection of all σ - algebras containing all open subsets. Since the average of σ - algebras over a common ground set again is a σ - algebra over this basic set and the power set always both σ -algebra and contains all open subsets, the existence of smallest σ - algebra is guaranteed.
• A Borel σ - algebra thus makes it possible to equip a topological space in a canonical way with the additional structure of a measurement space. In view of this structure, the space is then Borel space.
• The class of Borel sets is a subclass of the class of Suslin or analytical quantities.

Nomenclature for certain Borel sets

• In the literature, the following introduced by Felix Hausdorff name for some simple classes of Borel sets is enforced:

• In the descriptive set theory is called the open sets as sets, the sets as sets, the sets as sets, etc. complements of sets are called sets; such as the sets are exactly the sets.

Examples

The Borel σ - algebra on separable metric spaces

Given a separable metric space. The open balls generate a topology as a basis, this is called the metric topology generated. Every open set is due to the separability (which in the metric case, the second axiom of countability is equivalent ) to write it as a countable union of open balls. The smallest algebra, which contains the open spheres, therefore, contains all the open sets and is thus equal to the Borel algebra.

In the special case and the Euclidean metric is discussed in more detail in the following sections.

The Borel σ - algebra on the real numbers

The set of real numbers is usually equipped with the topology is defined by the open intervals rational endpoints. Although considered in individual cases, other topologies, this applies to as the canonical topology and the Borel σ - algebra derived from it is referred to simply as the Borel σ - algebra on. It contains (due to the seclusion of a σ - algebra with respect to the complementation ) except the open and the closed intervals.

The Borel σ - algebra of does not contain all subsets of. It can be shown even that the Borel σ - algebra of the same cardinality is closed, while the set of all subsets of features as a true greater power.

In measure theory showing that all Borel sets are Lebesgue measurable. However, the reverse statement does not apply; there Lebesgue measurable quantities that are not Borel quantities. These quantities, however, differ only by a quantity of Lebesgue measure 0 of a Borel set.

The Borel σ - algebra on finite dimensional real vector spaces

In the finite-dimensional vector spaces, the canonical topology with rational coordinates and spanned by the -dimensional blocks. It is simultaneously the - fold product topology of the canonical topology. The generated by it Borel σ - algebra is called analogous to one-dimensional case to the Borel σ - algebra.

In this way is also elegantly explains the Borel σ - algebra of complex numbers: one simply uses the Vektorraumisomorphie between and.

Application

The amount along with the borel between σ -algebra is a measurable space and the Borel measure as such is based. All elements of the borel σ - algebra between (which itself amounts ) are called Borel - measurable; only these are assigned by a Borel measure values ​​.