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-algebra, known as the Borel algebra or Borel σ-algebra. The Borel algebra on X is the smallest σ-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-compact spaces, but can be different in more pathological spaces.

Generating the Borel algebra

In the case 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

• $T_\sigma \quad$ be all countable unions of elements of T
• $T_\delta \quad$ 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,

  G⁰ = the collection of open subsets of X.

• If i is not a limit ordinal, then i has an immediately preceding ordinal i − 1. Let

  Gⁱ = [G^{i − 1}]_δσ.

• If i is a limit ordinal, set

  $G^i = \bigcup_{j < i} G^j.$

We now claim that the Borel algebra is G^ω₁, where ω₁ 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 fact, note that any open set in a metric space is the union of an increasing sequence of closed sets. In particular, it is easy to show that 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.

Note that for each Borel set B, there is some countable ordinal α_B such that B can be obtained by iterating the operation over α_B. However, as B varies over all Borel sets, α_B will vary over all the countable ordinals, and thus the first ordinal at which all the Borel sets are obtained is ω₁, the first uncountable ordinal.

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-algebra on R which 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 power of the continuum. So, the total number of Borel sets is less than or equal to $\aleph_1 \times 2 ^ {\aleph_0}\, = 2^{\aleph_0}\,$.

Standard Borel spaces and Kuratowski theorems

The following is one of a number of theorems of Kuratowski on Borel spaces: A Borel space is just another name for a set equipped with a distinguished σ-algebra; by extension elements of the distinguished σ-algebra are called Borel sets. Borel spaces form a category in which the maps are Borel measurable mappings between Borel spaces, where

$f:X \rightarrow Y$

is Borel measurable means that f ^{− 1}(B) is Borel in X for any Borel subset B of Y.

Theorem. Let X be a Polish space, that is, a topological space such that there is a metric d on X which defines the topology of X and which makes X a complete separable metric space. Then X as a Borel space is isomorphic to one of (1) R, (2) Z or (3) a finite space.

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

A standard Borel space is the Borel space associated to a Polish space.

Note that any standard Borel space is defined (up to isomorphism) by its cardinality^[1], 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 which 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 which is non-Borel, due to Lusin, is described below. In contrast, an example of a non-measurable set cannot be exhibited, though its existence can be proved.

Every irrational number has a unique representation by a 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. In fact, it is analytic, and complete in the class of analytic sets. For more details see descriptive set theory and the book by Kechris, especially Exercise (27.2) on page 209, Definition (22.9) on page 169, and Exercise (3.4)(ii) on page 14.

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 choice axiom), not an explicit example.

See also

• Baire set
• Cylindrical σ-algebra
• Polish space
• Descriptive set theory

References

An excellent exposition of the machinery of Polish topology is given in Chapter 3 of the following reference:

• William Arveson, An Invitation to C*-algebras, Springer-Verlag, 1981

• Richard Dudley, Real Analysis and Probability. Wadsworth, Brooks and Cole, 1989

• Paul Halmos, Measure Theory, D.van Nostrand Co., 1950

• Halsey Royden, Real Analysis, Prentice Hall, 1988

• Alexander S. Kechris, Classical Descriptive Set Theory, Springer-Verlag, 1995 (Graduate texts in Math., vol. 156)

1. ^ Srivastava, S.M. (1991), A Course on Borel Sets, Springer Verlag, ISBN 0-387-98412-7