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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 open sets and closed sets must also be defined on all Borel sets. 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, the Borel sets are defined using compact sets and their complements rather than closed and open sets. These two definitions are equivalent for most typical spaces, including any locally compact, separable metric space (or more generally any σ-compact space), but are different for certain pathological spaces.

1 Generating the Borel algebra
- 1.1 Example
2 Standard Borel spaces and Kuratowski theorems
3 See also
4 References

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,

G 0

= 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] 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. (Note: for any fixed Borel set, we only have to iterate a countable number of times, but as we vary across all Borel sets, this countable number of times is arbitrarily large and approaches 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; moreover if m is an uncountable limit ordinal, G^m is closed under countable unions.

This alternate definition is useful for some set-theoretic considerations, but the minimalist definition is preferred by analysts.

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.

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.

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

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Borel set

Contents

Generating the Borel algebra

Example

Standard Borel spaces and Kuratowski theorems

See also

References