In mathematics, the Borel algebra (or Borel σ-algebra) on a topological space X is a σ-algebra of subsets of X associated to the topology of X. In the mathematics literature, there are at least two inequivalent definitions of this σ-algebra: Euclid, detail from The School of Athens by Raphael. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, a σ-algebra (or σ-field) X over a set S is a family of subsets of S which is closed under countable set operations; σ-algebras are mainly used in order to define measures on S. The concept is important in mathematical analysis and probability theory. ...

• The minimal σ-algebra containing the open sets.
• The minimal σ-algebra containing the compact sets.

Here, the minimal σ-algebra containing a collection T of subsets of X is the smallest σ-algebra containing T. The existence and uniqueness of the minimal σ-algebra is shown by noting that the intersection of all σ-algebras containing T is itself a σ-algebra containing T. In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... In mathematics, a compact set is a set of points in a topological space such that every one of its (possibly infinite) open covers has a finite subcover. ... A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, a set A is a subset of a set B, if A is contained inside B. The relationship of one set being a subset of another is called inclusion. ... In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements. ...

The elements of the Borel algebra are called Borel sets, and a subset of X which is a Borel set is called a Borel subset.

In general topological spaces, even locally compact ones, the two structures can be different, although this phenomenon is generally considered to be pathological in mathematical analysis. Indeed, the two structures are identical whenever the topological space is a locally compact separable metric space. In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. ... In topology and related areas of mathematics a topological space is called separable if it contains a countable dense subset; that is, a set with a countable number of elements whose closure is the entire space. ... In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ...

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

• be all countable unions of elements of T
• be all countable intersections of elements of T
• 

Define by transfinite induction a sequence G^m, where m is an ordinal number, in the following manner: Transfinite induction is the proof technique of mathematical induction when applied to (large) well-ordered sets, for instance to sets of ordinals or cardinals, or even to the class of all ordinals. ... Commonly, ordinal numbers, or ordinals for short, are numbers used to denote the position in an ordered sequence: first, second, third, fourth, etc. ...

• 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

• If i is a limit ordinal, set

We now claim that the Borel algebra is G^m for the first uncountable ordinal number m, that is, the Borel algebra can be generated from the class of open sets by iterating the operation A limit ordinal is an ordinal number which is not a successor ordinal. ...

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. Probability theory is the mathematical study of probability. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... In mathematics, the Borel algebra is the smallest σ-algebra on the real numbers R containing the intervals, and the Borel measure is the measure on this σ-algebra which gives to the interval [a, b] the measure b − a (where a < b). ... In mathematics, a probability space or probability measure is a set S, together with a σ-algebra X on S and a measure P on that σ-algebra such that P(S) = 1. ... In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ... In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ...

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 σ-algebra. Borel spaces form a category in which the maps are Borel measurable mappings between Borel spaces, where Kazimierz Kuratowski (born February 2, 1896, Warsaw, died June 18, 1980, Warsaw) was a Polish mathematician. ... In mathematics, categories allow one to formalize notions involving abstract structure and processes which preserve structure. ...

f:X → 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. In mathematics, a Polish space is a separable completely metrisable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset. ... See: International System of Units, colloquially called the Metric System, and also metrication. ...

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. In mathematical logic and descriptive set theory, the analytical hierarchy is a second-order analogue of the arithmetical hierarchy. ...

See also

• Baire set

asdfasfasdf In mathematics, Baire set may mean: set having the property of Baire Baire set (general topology), an element of the smallest sigma-algebra containing all compact Gδ sets. ...

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