Inmathematics, theBorel sets of atopological space are a particular class of "well-behaved" subsets of that space. For example, whereas an arbitrary subset of thereal numbers might fail to beLebesgue measurable, every Borel set of reals isuniversally measurable. Which sets are Borel can be specified in a number of equivalent ways. Borel sets are named afterÉmile Borel.

The most usual definition goes through the notion of aσ-algebra, which is a collection of subsets of a topological space${\displaystyle X}$ that contains both the empty set and the entire set${\displaystyle X}$, and is closed undercountable union and countable intersection.^[a]

Then we can define theBorel σ-algebra over${\displaystyle X}$ to be the smallest σ-algebra containing all open sets of${\displaystyle X}$.^[b] A Borel subset of${\displaystyle X}$ is then simply an element of this σ-algebra.

Borel sets are important inmeasure theory, since any measure defined on the open sets of a space, or on the closed sets of a space, must also be defined on all Borel sets of that space. Any measure defined on the Borel sets is called aBorel measure. Borel sets and the associatedBorel hierarchy also play a fundamental role indescriptive set theory.

In some contexts, Borel sets are defined to be generated by thecompact sets of the topological space, rather than the open sets. The two definitions are equivalent for manywell-behaved spaces, including allHausdorffσ-compact spaces, but can be different in morepathological spaces.

In the case that${\displaystyle X}$ is ametric space, the Borel algebra in the first sense may be describedgeneratively as follows.

For a collection${\displaystyle T}$ of subsets of${\displaystyle X}$ (that is, for any subset of thepower set${\displaystyle \mathcal{P}}$${\displaystyle (X)}$ of${\displaystyle X}$), let

• ${\displaystyle T_{\sigma }}$ be all countable unions of elements of${\displaystyle T}$
• ${\displaystyle T_{\delta }}$ be all countable intersections of elements of${\displaystyle T}$
• ${\displaystyle T_{\delta \sigma }=(T_{\delta }{)}_{\sigma }.}$

Now define bytransfinite induction a sequence${\displaystyle G^m}$, where${\displaystyle m}$ is anordinal number, in the following manner:

• For the base case of the definition, let${\displaystyle G^0}$ be the collection of open subsets of${\displaystyle X}$.
• If${\displaystyle i}$ is not alimit ordinal, then${\displaystyle i}$ has an immediately preceding ordinal${\displaystyle i-1}$. Let$${\displaystyle G^i=[G^{i-1}{]}_{\delta \sigma }.}$$
• If${\displaystyle i}$ is a limit ordinal, set

The claim is that the Borel algebra is${\displaystyle G^{{\omega }_1}}$, where${\displaystyle {\omega }_1}$ is thefirst uncountable ordinal number. That is, the Borel algebra can begenerated from the class of open sets by iterating the operation$${\displaystyle G\mapsto G_{\delta \sigma }}$$to the first uncountable ordinal.

To prove this claim, any open set in a metric space is the union of an increasing sequence of closed sets. In particular, complementation of sets maps${\displaystyle G^m}$ into itself for any limit ordinal${\displaystyle m}$; moreover if${\displaystyle m}$ is an uncountable limit ordinal,${\displaystyle G^m}$ is closed under countable unions.

For each Borel set${\displaystyle B}$, there is some countable ordinal${\displaystyle {\alpha }_B}$ such that${\displaystyle B}$ can be obtained by iterating the operation over${\displaystyle {\alpha }_B}$. However, as${\displaystyle B}$ varies over all Borel sets,${\displaystyle {\alpha }_B}$ will vary over all the countable ordinals, and thus the first ordinal at which all the Borel sets are obtained is${\displaystyle {\omega }_1}$, the first uncountable ordinal.

The resulting sequence of sets is termed theBorel hierarchy.

An important example, especially in thetheory of probability, is the Borel algebra on the set ofreal numbers. It is the algebra on which theBorel measure is defined. Given areal random variable defined on aprobability space, itsprobability distribution is by definition also a measure on the Borel algebra.

The Borel algebra on the reals is the smallest σ-algebra on${\displaystyle \mathbb{R}}$ that contains all theintervals.

In the construction by transfinite induction, it can be shown that, in each step, thenumber of sets is, at most, thecardinality of the continuum. So, the total number of Borel sets is less than or equal to$${\displaystyle {\mathrm{\aleph }}_1\cdot 2^{{\mathrm{\aleph }}_0}=2^{{\mathrm{\aleph }}_0}.}$$

In fact, the cardinality of the collection of Borel sets is equal to that of the continuum (compare to the number ofLebesgue measurable sets that exist, which is strictly larger and equal to${\displaystyle 2^{2^{{\mathrm{\aleph }}_0}}}$).

Let${\displaystyle X}$ be a topological space. TheBorel space associated to${\displaystyle X}$ is the pair${\displaystyle (X,\mathcal{B})}$, where${\displaystyle \mathcal{B}}$ is the σ-algebra of Borel sets of${\displaystyle X}$.

George Mackey defined a Borel space somewhat differently, writing that it is "a set together with a distinguished σ-field of subsets called its Borel sets."^[1] However, modern usage is to call the distinguished sub-algebra themeasurable sets and such spacesmeasurable spaces. The reason for this distinction is that the Borel sets are the σ-algebra generated byopen sets (of a topological space), whereas Mackey's definition refers to a set equipped with anarbitrary σ-algebra. There exist measurable spaces that are not Borel spaces, for any choice of topology on the underlying space.^[2]

Measurable spaces form acategory in which themorphisms aremeasurable functions between measurable spaces. A function${\displaystyle f:X\to Y}$ ismeasurable if itpulls back measurable sets, i.e., for all measurable sets${\displaystyle B}$ in${\displaystyle Y}$, the set${\displaystyle f^{-1}(B)}$ is measurable in${\displaystyle X}$.

Theorem. Let${\displaystyle X}$ be aPolish space, that is, a topological space such that there is ametric${\displaystyle d}$ on${\displaystyle X}$ that defines the topology of${\displaystyle X}$ and that makes${\displaystyle X}$ a completeseparable metric space. Then${\displaystyle X}$ as a Borel space isisomorphic to one of

1. ${\displaystyle \mathbb{R}}$,
2. ${\displaystyle \mathbb{Z}}$,
3. a finite space.

(This result is reminiscent ofMaharam's theorem.)

Considered as Borel spaces, the real line${\displaystyle \mathbb{R}}$, the union of${\displaystyle \mathbb{R}}$ with a countable set, and${\displaystyle {\mathbb{R}}^n}$ are isomorphic.

Astandard Borel space is the Borel space associated to aPolish space. A standard Borel space is characterized up to isomorphism by its cardinality,^[3] and any uncountable standard Borel space has the cardinality of the continuum.

For subsets of Polish spaces, Borel sets can be characterized as those sets that are the ranges of continuous injective maps defined on Polish spaces. Note however, that the range of a continuous noninjective map may fail to be Borel. Seeanalytic set.

Everyprobability measure on a standard Borel space turns it into astandard probability space.

An example of a subset of the reals that is non-Borel, due toLusin,^[4] is described below. In contrast, an example of anon-measurable set cannot be exhibited, although the existence of such a set is implied, for example, by theaxiom of choice.

Everyirrational number has a unique representation by an infinitesimple continued fraction

${\displaystyle x=a_0+\frac{1}{a_1+\frac{1}{a_2+\frac{1}{a_3+\frac{1}{\ddots }}}}}$

where${\displaystyle a_0}$ is someinteger and all the other numbers${\displaystyle a_k}$ arepositive integers. Let${\displaystyle A}$ be the set of all irrational numbers that correspond to sequences${\displaystyle (a_0,a_1,\dots )}$ with the following property: there exists an infinitesubsequence${\displaystyle (a_{k_0},a_{k_1},\dots )}$ such that each element is adivisor of the next element. This set${\displaystyle A}$ is not Borel. However, it isanalytic (all Borel sets are also analytic), and complete in the class of analytic sets. For more details seedescriptive set theory and the book byA. S. Kechris (see References), especially Exercise (27.2) on page 209, Definition (22.9) on page 169, Exercise (3.4)(ii) on page 14, and on page 196.

It's important to note, that whileZermelo–Fraenkel axioms (ZF) are sufficient to formalize the construction of${\displaystyle A}$, it cannot be proven in ZF alone that${\displaystyle A}$ is non-Borel. In fact, it is consistent with ZF that${\displaystyle \mathbb{R}}$ is a countable union of countable sets,^[5] so that any subset of${\displaystyle \mathbb{R}}$ is a Borel set.

Another non-Borel set is an inverse image${\displaystyle f^{-1}[0]}$ of aninfinite parity function${\displaystyle f:\{0,1{\}}^{\omega }\to \{0,1\}}$. However, this is a proof of existence (via the axiom of choice), not an explicit example.

According toPaul Halmos,^[6] a subset of alocally compact Hausdorff topological space is called aBorel set if it belongs to the smallestσ-ring containing all compact sets.

Norberg and Vervaat^[7] redefine the Borel algebra of a topological space${\displaystyle X}$ as the${\displaystyle \sigma }$-algebra generated by its open subsets and its compactsaturated subsets. This definition is well-suited for applications in the case where${\displaystyle X}$ is not Hausdorff. It coincides with the usual definition if${\displaystyle X}$ issecond countable or if every compact saturated subset is closed (which is the case in particular if${\displaystyle X}$ is Hausdorff).

• Borel hierarchy – Mathematical logic hierarchy
• Borel isomorphism
• Baire set
• Cylindrical σ-algebra
• Descriptive set theory – Subfield of mathematical logic
• Polish space – Concept in topology

• William Arveson,An Invitation to C*-algebras, Springer-Verlag, 1981. (See Chapter 3 for an excellent exposition ofPolish topology)
• Richard Dudley, Real Analysis and Probability. Wadsworth, Brooks and Cole, 1989
• Halmos, Paul R. (1950).Measure theory. D. van Nostrand Co. See especially Sect. 51 "Borel sets and Baire sets".
• Halsey Royden,Real Analysis, Prentice Hall, 1988
• Alexander S. Kechris,Classical Descriptive Set Theory, Springer-Verlag, 1995 (Graduate texts in Math., vol. 156)

• "Borel set",Encyclopedia of Mathematics,EMS Press, 2001 [1994]
• Formal definition of Borel Sets in theMizar system, and thelist of theoremsArchived 2020-06-01 at theWayback Machine that have been formally proved about it.
• Weisstein, Eric W."Borel Set".MathWorld.

• Topology
• Descriptive set theory

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