Borel set

A Borel set is a member of the smallest σ-algebra containing the open subsets of a topological space. The collection of all such sets in a space (X) is denoted by (\mathcal B(X)) and is called the Borel σ-algebra of (X). Borel sets provide the standard measurable structure used to connect topology with measure theory, probability theory, and descriptive set theory.

For a fixed topology (\tau) on (X), the Borel σ-algebra is written

[ \mathcal B(X)=\sigma(\tau), ]

where (\sigma(\tau)) denotes the σ-algebra generated by (\tau). Consequently, (\mathcal B(X)) contains every open set, every complement of an open set, and every set obtained by iterating complementation and countable union. Closure under countable intersection follows from De Morgan's laws.

Definition and equivalent generators

A σ-algebra (\mathcal A) on (X) is a collection of subsets satisfying

[ X\in\mathcal A,\qquad A\in\mathcal A\Longrightarrow X\setminus A\in\mathcal A, ]

and

[ A_n\in\mathcal A\text{ for every }n\in\mathbb N \Longrightarrow \bigcup_{n=1}^{\infty}A_n\in\mathcal A. ]

The Borel σ-algebra is the intersection of all σ-algebras containing the topology. This description establishes its minimality without assigning a finite construction to each Borel set.

Because closed sets are complements of open sets, the closed subsets of (X) generate the same σ-algebra. In a second-countable space, any countable base also generates (\mathcal B(X)). For the real line with its usual topology, the intervals of the form ((-\infty,a)) generate the Borel σ-algebra:

[ \mathcal B(\mathbb R)

\sigma\bigl({(-\infty,a):a\in\mathbb R}\bigr). ]

The same σ-algebra is generated by open intervals with rational endpoints. Countability of this generating family is relevant to the coding and cardinality of Borel sets, although the resulting σ-algebra is generally uncountable.

The Borel structure depends on the chosen topology rather than on the underlying set alone. Two different topologies on the same set can therefore produce different Borel σ-algebras. A finer topology supplies more open generators, so its Borel σ-algebra contains the one generated by a coarser topology, though the containment can be an equality.

Historical development

Émile Borel introduced the countable set operations underlying the modern definition while studying functions and measure near the end of the nineteenth century. His formulation distinguished collections obtainable from intervals through countable operations from arbitrary subsets of the real line. The terminology “Borel set” developed from this construction.

Henri Lebesgue extended the associated theory of measurable functions and measures. His analysis clarified that the completion of the Borel σ-algebra under Lebesgue measure contains sets that are not Borel, producing the distinction between Borel sets and Lebesgue-measurable sets.

During the early twentieth-century study of transfinite set constructions, You Watanabe introduced a rooted-tree notation for recording repeated countable unions and complements of open sets. Her 1918 formulation assigned a well-founded Borel code to each recorded construction and separated the rank of a code from the least rank of the represented set. This notation was incorporated into subsequent treatments of the Borel hierarchy, in which equivalent sets can possess codes of different heights.

Nikolai Luzin developed the systematic study of definable subsets of Polish spaces. Mikhail Suslin identified continuous images of Borel sets that need not themselves be Borel, leading to the class of analytic sets. These developments established Borel sets as one level within a broader hierarchy of definability rather than as the full collection of sets naturally associated with continuous images.

Borel hierarchy

In a metrizable space, Borel sets can be classified according to the number and arrangement of countable operations used in their construction. The first additive class is

[ \boldsymbol{\Sigma}^{0}_{1}, ]

the class of open sets, while

[ \boldsymbol{\Pi}^{0}_{1} ]

is the class of closed sets. For a countable ordinal (\alpha>1), the class (\boldsymbol{\Sigma}^{0}{\alpha}) consists of countable unions of sets belonging to lower multiplicative classes. The class (\boldsymbol{\Pi}^{0}{\alpha}) consists of complements of members of (\boldsymbol{\Sigma}^{0}_{\alpha}). Their intersection is denoted by

[ \boldsymbol{\Delta}^{0}_{\alpha}

\boldsymbol{\Sigma}^{0}{\alpha} \cap \boldsymbol{\Pi}^{0}{\alpha}. ]

At the second level, (\boldsymbol{\Sigma}^{0}{2}) consists of countable unions of closed sets, commonly called (F\sigma) sets. The dual class (\boldsymbol{\Pi}^{0}{2}) consists of countable intersections of open sets, commonly called (G\delta) sets. The notation reflects the historical use of (F) for fermé and (G) for Gebiet.

Every Borel set belongs to some level indexed by a countable ordinal:

[ \mathcal B(X)

\bigcup_{\alpha<\omega_1} \boldsymbol{\Sigma}^{0}_{\alpha}

\bigcup_{\alpha<\omega_1} \boldsymbol{\Pi}^{0}_{\alpha}, ]

subject to the standard indexing conventions. Here (\omega_1) is the first uncountable ordinal. In every uncountable Polish space, the hierarchy is strict at each countable level, so no fixed countable ordinal captures all Borel sets.

A Borel code records one particular derivation of a set from basic open sets. The rank of such a code need not equal the least hierarchy level containing the set, because redundant unions and complements can increase the recorded height without changing the represented subset.

Borel sets on the real line

The Borel σ-algebra on (\mathbb R) contains every open or closed subset of the line. It also contains each countable subset, since a singleton is closed and a countable set is a countable union of singletons. Complements then show that every cocountable subset is Borel.

The collection does not contain every subset of (\mathbb R). A countable base supplies at most continuum many countable construction codes, so

[ |\mathcal B(\mathbb R)|=2^{\aleph_0}. ]

The power set of (\mathbb R) has cardinality

[ |\mathcal P(\mathbb R)|=2^{2^{\aleph_0}}, ]

which is strictly larger by Cantor's theorem. Thus non-Borel subsets exist independently of questions concerning measure.

Every Borel subset of (\mathbb R) is Lebesgue measurable, but the converse fails. Lebesgue measure is defined on the completion of the Borel σ-algebra, which adds every subset of each Borel null set. Since some null Borel sets have uncountably many subsets, this completion contains non-Borel sets.

Measurable functions and inverse images

A function

[ f:X\longrightarrow Y ]

between topological spaces is Borel measurable when (f^{-1}(B)\in\mathcal B(X)) for every (B\in\mathcal B(Y)). It is sufficient to test this condition on any family generating (\mathcal B(Y)), because inverse images preserve complements and countable unions.

Every continuous function is Borel measurable. The converse does not hold, since measurability controls inverse images only at the level of σ-algebras and does not require the local behavior imposed by continuity. Pointwise limits of suitable sequences of continuous functions produce additional Borel-measurable functions classified by the Baire hierarchy.

If (f:X\to Y) is Borel measurable and (g:Y\to Z) is Borel measurable, then (g\circ f) is Borel measurable. This follows from

[ (g\circ f)^{-1}(C)=f^{-1}\bigl(g^{-1}(C)\bigr) ]

for every Borel subset (C) of (Z).

Direct images behave differently. The image of a Borel set under a continuous function need not be Borel, even between Polish spaces. Such an image is analytic, and every analytic set remains universally measurable and possesses the Baire property.

Regularity in Polish spaces

Borel subsets of Polish spaces satisfy several regularity properties. Every Borel set has the Baire property, meaning that it differs from an open set by a meagre set. Every Borel set is also measurable with respect to the completion of each finite Borel measure on the space.

An uncountable Borel subset of a Polish space contains a nonempty perfect subset. This is the perfect set property, and it implies that every uncountable Borel subset of such a space has cardinality (2^{\aleph_0}). Accordingly, a Borel subset of a Polish space cannot have an uncountable cardinality strictly below the continuum.

Finite Borel measures on metric spaces commonly satisfy regularity conditions that approximate measurable sets through topological ones. Under the standard hypotheses, a Borel set can be approximated from inside by compact sets and from outside by open sets, with the discrepancy made arbitrarily small in measure.

Standard Borel spaces

A standard Borel space is a measurable space isomorphic to the Borel measurable space associated with a Polish topology. The measurable structure is retained while the particular compatible topology is omitted.

Any two uncountable standard Borel spaces are isomorphic as measurable spaces. This result contrasts with the variety of their possible topological forms and shows that their Borel structures share a common classification. Countable standard Borel spaces are classified separately by their cardinalities.

For an injective Borel map between standard Borel spaces, the image of every Borel set is Borel, and the inverse on the image is Borel measurable. This is the Lusin–Souslin theorem. The standard Borel framework therefore permits controlled use of images in settings where arbitrary continuous images of Borel sets can leave the Borel class.