Borel set

In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are named after Émile Borel.

For a topological space X, the collection of all Borel sets on X forms a σしぐま-algebra, known as the Borel algebra or Borel σしぐま-algebra. The Borel algebra on X is the smallest σしぐま-algebra containing all open sets (or, equivalently, all closed sets).

Borel sets are important in measure 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 a Borel measure. Borel sets and the associated Borel hierarchy also play a fundamental role in descriptive set theory.

In some contexts, Borel sets are defined to be generated by the compact sets of the topological space, rather than the open sets. The two definitions are equivalent for many well-behaved spaces, including all Hausdorff σしぐま-compact spaces, but can be different in more pathological spaces.

Generating the Borel algebra

In the case that X is a metric space, the Borel algebra in the first sense may be described generatively as follows.

For a collection T of subsets of X (that is, for any subset of the power set P(X) of X), let

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

Now define by transfinite induction a sequence G^m, where m is an ordinal number, in the following manner:

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

The claim is that the Borel algebra is G^{ωおめが₁}, where ωおめが₁ is the first uncountable ordinal number. That is, the Borel algebra can be generated from the class of open sets by iterating the operation $G\mapsto G_{\delta \sigma }.$ 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 G^m into itself for any limit ordinal m; moreover if m is an uncountable limit ordinal, G^m is closed under countable unions.

For each Borel set B, there is some countable ordinal αあるふぁ_B such that B can be obtained by iterating the operation over αあるふぁ_B. However, as B varies over all Borel sets, αあるふぁ_B will vary over all the countable ordinals, and thus the first ordinal at which all the Borel sets are obtained is ωおめが₁, the first uncountable ordinal.

The resulting sequence of sets is termed the Borel hierarchy.

Example

An important example, especially in the theory of probability, is the Borel algebra on the set of real numbers. It is the algebra on which the Borel measure is defined. Given a real random variable defined on a probability space, its probability distribution is by definition also a measure on the Borel algebra.

The Borel algebra on the reals is the smallest σしぐま-algebra on R that contains all the intervals.

In the construction by transfinite induction, it can be shown that, in each step, the number of sets is, at most, the cardinality of the continuum. So, the total number of Borel sets is less than or equal to $\aleph _{1}\cdot 2^{\aleph _{0}}\,=2^{\aleph _{0}}.$

In fact, the cardinality of the collection of Borel sets is equal to that of the continuum (compare to the number of Lebesgue measurable sets that exist, which is strictly larger and equal to $2^{2^{\aleph _{0}}}$ ).

Standard Borel spaces and Kuratowski theorems

Let X be a topological space. The Borel space associated to X is the pair (X,B), where B is the σしぐま-algebra of Borel sets of X.

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

Measurable spaces form a category in which the morphisms are measurable functions between measurable spaces. A function $f:X\rightarrow Y$ is measurable if it pulls back measurable sets, i.e., for all measurable sets B in Y, the set $f^{-1}(B)$ is measurable in X.

Theorem. Let X be a Polish space, that is, a topological space such that there is a metric d on X that defines the topology of X and that makes X a complete separable metric space. Then X as a Borel space is isomorphic to one of

R,
Z^{[clarification needed]},
a finite space.

(This result is reminiscent of Maharam's theorem.)

Considered as Borel spaces, the real line R, the union of R with a countable set, and Rⁿ are isomorphic.

A standard Borel space is the Borel space associated to a Polish 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. See analytic set.

Every probability measure on a standard Borel space turns it into a standard probability space.

Non-Borel sets

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

Every irrational number has a unique representation by an infinite continued fraction

x=a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{a_{3}+{\cfrac {1}{\ddots \,}}}}}}}}

where $a_{0}$ is some integer and all the other numbers $a_{k}$ are positive integers. Let $A$ be the set of all irrational numbers that correspond to sequences $(a_{0},a_{1},\dots )$ with the following property: there exists an infinite subsequence $(a_{k_{0}},a_{k_{1}},\dots )$ such that each element is a divisor of the next element. This set $A$ is not Borel. However, it is analytic (all Borel sets are also analytic), and complete in the class of analytic sets. For more details see descriptive set theory and the book by A. 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 while Zermelo–Fraenkel axioms (ZF) are sufficient to formalize the construction of $A$ , it cannot be proven in ZF alone that $A$ is non-Borel. In fact, it is consistent with ZF that $\mathbb {R}$ is a countable union of countable sets,^[5] so that any subset of $\mathbb {R}$ is a Borel set.

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

Alternative non-equivalent definitions

According to Paul Halmos,^[6] a subset of a locally compact Hausdorff topological space is called a Borel set if it belongs to the smallest σしぐま-ring containing all compact sets.

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

Notes

^ Mackey, G.W. (1966), "Ergodic Theory and Virtual Groups", Math. Ann., 166 (3): 187–207, doi:10.1007/BF01361167, ISSN 0025-5831, S2CID 119738592
^ Jochen Wengenroth, Is every sigma-algebra the Borel algebra of a topology?
^ Srivastava, S.M. (1991), A Course on Borel Sets, Springer Verlag, ISBN 978-0-387-98412-4
^ Lusin, Nicolas (1927), "Sur les ensembles analytiques", Fundamenta Mathematicae (in French), 10: Sect. 62, pages 76–78, doi:10.4064/fm-10-1-1-95
^ Jech, Thomas (2008). The Axiom of Choice. Courier Corporation. p. 142.
^ (Halmos 1950, page 219)
^ Tommy Norberg and Wim Vervaat, Capacities on non-Hausdorff spaces, in: Probability and Lattices, in: CWI Tract, vol. 110, Math. Centrum Centrum Wisk. Inform., Amsterdam, 1997, pp. 133-150

References

William Arveson, An Invitation to C*-algebras, Springer-Verlag, 1981. (See Chapter 3 for an excellent exposition of Polish 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)

External links

"Borel set", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
Formal definition of Borel Sets in the Mizar system, and the list of theorems Archived 2020-06-01 at the Wayback Machine that have been formally proved about it.
Weisstein, Eric W. "Borel Set". MathWorld.

Lightface		Boldface
Σしぐま⁰ ₀ = Πぱい⁰ ₀ = Δでるた⁰ ₀ (sometimes the same as Δでるた⁰ ₁)		Σしぐま⁰ ₀ = Πぱい⁰ ₀ = Δでるた⁰ ₀ (if defined)
Δでるた⁰ ₁ = recursive		Δでるた⁰ ₁ = clopen
Σしぐま⁰ ₁ = recursively enumerable	Πぱい⁰ ₁ = co-recursively enumerable	Σしぐま⁰ ₁ = G = open	Πぱい⁰ ₁ = F = closed
Δでるた⁰ ₂		Δでるた⁰ ₂
Σしぐま⁰ ₂	Πぱい⁰ ₂	Σしぐま⁰ ₂ = F_σしぐま	Πぱい⁰ ₂ = G_δでるた
Δでるた⁰ ₃		Δでるた⁰ ₃
Σしぐま⁰ ₃	Πぱい⁰ ₃	Σしぐま⁰ ₃ = G_{δでるたσしぐま}	Πぱい⁰ ₃ = F_{σしぐまδでるた}
⋮		⋮
Σしぐま⁰ _<ωおめが = Πぱい⁰ _<ωおめが = Δでるた⁰ _<ωおめが = Σしぐま¹ ₀ = Πぱい¹ ₀ = Δでるた¹ ₀ = arithmetical		Σしぐま⁰ _<ωおめが = Πぱい⁰ _<ωおめが = Δでるた⁰ _<ωおめが = Σしぐま¹ ₀ = Πぱい¹ ₀ = Δでるた¹ ₀ = boldface arithmetical
⋮		⋮
Δでるた⁰ _{αあるふぁ} (αあるふぁ recursive)		Δでるた⁰ _{αあるふぁ} (αあるふぁ countable)
Σしぐま⁰ _{αあるふぁ}	Πぱい⁰ _{αあるふぁ}	Σしぐま⁰ _{αあるふぁ}	Πぱい⁰ _{αあるふぁ}
⋮		⋮
Σしぐま⁰ _{ωおめが^CK ₁} = Πぱい⁰ _{ωおめが^CK ₁} = Δでるた⁰ _{ωおめが^CK ₁} = Δでるた¹ ₁ = hyperarithmetical		Σしぐま⁰ _{ωおめが₁} = Πぱい⁰ _{ωおめが₁} = Δでるた⁰ _{ωおめが₁} = Δでるた¹ ₁ = B = Borel
Σしぐま¹ ₁ = lightface analytic	Πぱい¹ ₁ = lightface coanalytic	Σしぐま¹ ₁ = A = analytic	Πぱい¹ ₁ = CA = coanalytic
Δでるた¹ ₂		Δでるた¹ ₂
Σしぐま¹ ₂	Πぱい¹ ₂	Σしぐま¹ ₂ = PCA	Πぱい¹ ₂ = CPCA
Δでるた¹ ₃		Δでるた¹ ₃
Σしぐま¹ ₃	Πぱい¹ ₃	Σしぐま¹ ₃ = PCPCA	Πぱい¹ ₃ = CPCPCA
⋮		⋮
Σしぐま¹ _<ωおめが = Πぱい¹ _<ωおめが = Δでるた¹ _<ωおめが = Σしぐま² ₀ = Πぱい² ₀ = Δでるた² ₀ = analytical		Σしぐま¹ _<ωおめが = Πぱい¹ _<ωおめが = Δでるた¹ _<ωおめが = Σしぐま² ₀ = Πぱい² ₀ = Δでるた² ₀ = P = projective
⋮		⋮

[1] Mackey, G.W. (1966), "Ergodic Theory and Virtual Groups", Math. Ann., 166 (3): 187–207, doi:10.1007/BF01361167, ISSN 0025-5831, S2CID 119738592

[2] Jochen Wengenroth, Is every sigma-algebra the Borel algebra of a topology?

[3] Srivastava, S.M. (1991), A Course on Borel Sets, Springer Verlag, ISBN 978-0-387-98412-4

[4] Lusin, Nicolas (1927), "Sur les ensembles analytiques", Fundamenta Mathematicae (in French), 10: Sect. 62, pages 76–78, doi:10.4064/fm-10-1-1-95

[5] Jech, Thomas (2008). The Axiom of Choice. Courier Corporation. p. 142.

[6] (Halmos 1950, page 219)

[7] Tommy Norberg and Wim Vervaat, Capacities on non-Hausdorff spaces, in: Probability and Lattices, in: CWI Tract, vol. 110, Math. Centrum Centrum Wisk. Inform., Amsterdam, 1997, pp. 133-150

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