In
mathematicsMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
, a
Borel set is any set in a
topological spaceTopological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
that can be formed from
open setIn mathematics, more specifically point-set topology and metric topology, the notion of an open set provides a fundamental way to speak of distance in a topological space, without explicitly defining a metric on the space...
s (or, equivalently, from
closed setIn topology and related branches of mathematics, a closed set is a set whose complement is open.- Equivalent definitions of a closed set :In a topological space, a set is closed if and only if it coincides with its closure...
s) through the operations of countable
unionIn set theory, the union of a collection of sets is the set of all distinct elements in the collection. The union of a collection of sets gives a set .- Definition :A simple example:...
, countable
intersectionIn mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....
, and relative complement. Borel sets are named after
Émile BorelFélix Édouard Justin Émile Borel was a French mathematician[ - Université Lille Nord de France] and politician....
.
For a topological space
X, the collection of all Borel sets on
X forms a
σ-algebraIn mathematics, a σ-algebra over a set X is a nonempty collection Σ of subsets of X that is closed under complementation and countable unions of its members. It is a Boolean algebra, completed to include countably infinite operations...
, known as the
Borel algebra or
Borel σ-algebraIn mathematics, a σ-algebra over a set X is a nonempty collection Σ of subsets of X that is closed under complementation and countable unions of its members. It is a Boolean algebra, completed to include countably infinite operations...
. 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.
In
mathematicsMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
, a
Borel set is any set in a
topological spaceTopological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
that can be formed from
open setIn mathematics, more specifically point-set topology and metric topology, the notion of an open set provides a fundamental way to speak of distance in a topological space, without explicitly defining a metric on the space...
s (or, equivalently, from
closed setIn topology and related branches of mathematics, a closed set is a set whose complement is open.- Equivalent definitions of a closed set :In a topological space, a set is closed if and only if it coincides with its closure...
s) through the operations of countable
unionIn set theory, the union of a collection of sets is the set of all distinct elements in the collection. The union of a collection of sets gives a set .- Definition :A simple example:...
, countable
intersectionIn mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....
, and relative complement. Borel sets are named after
Émile BorelFélix Édouard Justin Émile Borel was a French mathematician[ - Université Lille Nord de France] and politician....
.
For a topological space
X, the collection of all Borel sets on
X forms a
σ-algebraIn mathematics, a σ-algebra over a set X is a nonempty collection Σ of subsets of X that is closed under complementation and countable unions of its members. It is a Boolean algebra, completed to include countably infinite operations...
, known as the
Borel algebra or
Borel σ-algebraIn mathematics, a σ-algebra over a set X is a nonempty collection Σ of subsets of X that is closed under complementation and countable unions of its members. It is a Boolean algebra, completed to include countably infinite operations...
. 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 measureIn mathematics, the Borel algebra is the smallest σ-algebra on the real numbers R containing theintervals, and the Borel measure is the measure on this σ-algebra which gives to the interval [a, b] the measure b − a.The Borel measure is not complete, which is...
. Borel sets and the associated
Borel hierarchyIn mathematical logic, the Borel hierarchy is a stratification of the Borel algebra generated by the open subsets of a Polish space; elements of this algebra are called Borel sets. Each Borel set is assigned a unique countable ordinal number called the rank of the Borel set...
also play a fundamental role in
descriptive set theoryIn mathematical logic, descriptive set theory is the study of certain classes of "well-behaved" subsets of the real line and other Polish spaces...
.
In some contexts, the Borel sets are defined using compact sets and their
complementsIn discrete mathematics and predominantly in set theory, a complement is a concept used in comparisons of sets to refer to the unique values of one set in relation to another...
rather than closed and open sets. These two definitions are equivalent for most typical spaces, including any locally compact,
separableIn mathematics a topological space is called separable if it contains a countable dense subset; that is, there exists a sequence of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence....
metric spaceIn mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
(or more generally any σ-compact space), but are different for certain
pathologicalIn mathematics, a pathological phenomenon is one whose properties are considered atypically bad or counterintuitive.Often, when the usefulness of a theorem is challenged by counterexamples, defenders of the theorem argue that the exceptions are 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 setIn mathematics, given a set S, the power set of S, written , P, ℘ or 2
S, is the set of all subsets of S. In axiomatic set theory In mathematics, given a set S, the power set (or powerset) of S, written , P(S), ℘(S) or 2
S, is the set of all subsets of S. In...
P(
X) of
X), let
- be all countable unions of elements of T
- be all countable intersections of elements of T
Now define by
transfinite inductionTransfinite induction is an extension of mathematical induction to well-ordered sets, for instance to sets of ordinals or cardinals.- Transfinite induction :...
a sequence
Gm, where
m is an
ordinal numberIn set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals...
, in the following manner:
- For the base case of the definition, = the collection of open subsets of X.
- If i is not a limit ordinal
A limit ordinal is an ordinal number which is neither zero nor a successor ordinal.Various equivalent ways to express this are:*It cannot be reached via the ordinal successor operation S; in precise terms, we say λ is a limit ordinal if and only if λ > 0 and for any β < λ,...
, 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
Gω1, where ω
1 is the
first uncountable ordinal numberIn mathematics, the first uncountable ordinal, traditionally denoted by ω1 or sometimes by Ω, is the smallest ordinal number that, considered as a set, is uncountable. It is the supremum of all countable ordinals...
. That is, the Borel algebra can be
generated from the class of open sets by iterating the operation
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
Gm into itself for any limit ordinal; moreover if
m is an uncountable limit ordinal,
Gm 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 probabilityProbability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
, is the Borel algebra on the set of
real numberIn mathematics, the real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339..., where the digits continue in some way; or, the real...
s. It is the algebra on which the
Borel measureIn mathematics, the Borel algebra is the smallest σ-algebra on the real numbers R containing theintervals, and the Borel measure is the measure on this σ-algebra which gives to the interval [a, b] the measure b − a.The Borel measure is not complete, which is...
is defined. Given a real random variable defined on a
probability spaceIn probability theory, the probability space, or probability triple, is a concept which serves as a rigorous mathematical ground for the conventional idea of randomness...
, its
probability distributionIn probability theory and statistics, a probability distribution identifies either the probability of each value of an unidentified random variable , or the probability of the value falling within a particular interval...
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
intervalsIn mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...
.
In the construction by transfinite induction, it can be shown that, in each step, the
numberIn mathematics, the cardinality of a set is a measure of the "number of elements of the set". For example, the set A = {2, 4, 6} contains 3 elements, and therefore A has a cardinality of 3...
of sets is, at most, the power of the continuum. So, the total number of Borel sets is less than or equal to .
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
categoryIn mathematics, a category is an algebraic structure consisting of a collection of "objects", linked together by a collection of "arrows" that have two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. Objects and arrows may...
in which the maps are Borel measurable mappings between Borel spaces, where
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 spaceIn mathematics, a Polish space is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset. Polish spaces are so named because they were first extensively studied by Polish topologists and logicians —...
, that is, a topological space such that there is a
metricIn mathematics, a metric or distance function is a function which defines a distance between elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set but not all topologies can be generated by 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 setIn descriptive set theory, a subset of a Polish space is an analytic set if it is a continuous image of a Polish space. These sets were first defined by and his student .- Definition :There are several equivalent definitions of analytic set...
.
Every probability measure on a standard Borel space turns it into a
standard probability spaceIn probability theory, a standard probability space is a probability space satisfying certain assumptions introduced by Vladimir Rokhlin in 1940...
.
See also
- Baire set
In mathematics, more specifically in measure theory, the notion of a Baire set is important in the understanding of particular relations between measure theory and topology. In particular, an understanding of Baire sets aids in intuition when one deals with measures on non-metrizable topological...
- Cylindrical σ-algebra
- Polish space
In mathematics, a Polish space is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset. Polish spaces are so named because they were first extensively studied by Polish topologists and logicians —...
- Descriptive set theory
In mathematical logic, descriptive set theory is the study of certain classes of "well-behaved" subsets of the real line and other Polish spaces...
- Non-Borel set
In mathematics, a non-Borel set is a set that cannot be obtained from simple sets by taking complements and at most countable unions and intersections. Only sets of real numbers are considered in this article. Accordingly, by simple sets one may mean just intervals...