In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a Borel set is any set in a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
that can be formed from
open set
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suf ...
s (or, equivalently, from
closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a cl ...
s) through the operations of
countable
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...
union
Union commonly refers to:
* Trade union, an organization of workers
* Union (set theory), in mathematics, a fundamental operation on sets
Union may also refer to:
Arts and entertainment
Music
* Union (band), an American rock group
** ''Un ...
, countable
intersection
In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their i ...
, and
relative complement
In set theory, the complement of a set , often denoted by (or ), is the set of elements not in .
When all sets in the universe, i.e. all sets under consideration, are considered to be members of a given set , the absolute complement of is th ...
. Borel sets are named after
Émile Borel
Félix Édouard Justin Émile Borel (; 7 January 1871 – 3 February 1956) was a French mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Math ...
.
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
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures ( length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simil ...
, 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
In mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all Borel sets). Some authors require additional restrictions on the measure, as described below.
F ...
. Borel sets and the associated
Borel hierarchy In 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 ...
also play a fundamental role in
descriptive set theory
In mathematical logic, descriptive set theory (DST) is the study of certain classes of "well-behaved" subsets of the real line and other Polish spaces. As well as being one of the primary areas of research in set theory, it has applications to ot ...
.
In some contexts, Borel sets are defined to be generated by the
compact set
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i ...
s of the topological space, rather than the open sets. The two definitions are equivalent for many
well-behaved
In mathematics, when a mathematical phenomenon runs counter to some intuition, then the phenomenon is sometimes called pathological. On the other hand, if a phenomenon does not run counter to intuition,
it is sometimes called well-behaved. Th ...
spaces, including all
Hausdorff σ-compact space
In mathematics, a topological space is said to be σ-compact if it is the union of countably many compact subspaces.
A space is said to be σ-locally compact if it is both σ-compact and locally compact.
Properties and examples
* Every compact ...
s, but can be different in more
pathological
Pathology is the study of the causal, causes and effects of disease or injury. The word ''pathology'' also refers to the study of disease in general, incorporating a wide range of biology research fields and medical practices. However, when us ...
spaces.
Generating the Borel algebra
In the case that ''X'' is a
metric space
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
, 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
In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is po ...
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 induction
Transfinite induction is an extension of mathematical induction to well-ordered sets, for example to sets of ordinal numbers or cardinal numbers. Its correctness is a theorem of ZFC.
Induction by cases
Let P(\alpha) be a property defined for a ...
a sequence ''G
m'', where ''m'' is an
ordinal number
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets.
A finite set can be enumerated by successively labeling each element with the least n ...
, in the following manner:
* For the base case of the definition, let
be the collection of open subsets of ''X''.
* If ''i'' is not a
limit ordinal
In set theory, a limit ordinal is an ordinal number that is neither zero nor a successor ordinal. Alternatively, an ordinal λ is a limit ordinal if there is an ordinal less than λ, and whenever β is an ordinal less than λ, then there exists an ...
, then ''i'' has an immediately preceding ordinal ''i'' − 1. Let
* If ''i'' is a limit ordinal, set
The claim is that the Borel algebra is ''G''
ω1, where ω
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
to the first uncountable ordinal.
To prove this claim, note that 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.
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 ''ω''
1, the first uncountable ordinal.
Example
An important example, especially in the
theory of probability
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
, is the Borel algebra on the set of
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s. It is the algebra on which the
Borel measure
In mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all Borel sets). Some authors require additional restrictions on the measure, as described below.
F ...
is defined. Given a
real random variable defined on a
probability space
In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models t ...
, its
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
is by definition also a measure on the Borel algebra.
The Borel algebra on the reals is the smallest σ-algebra on R that contains all the
intervals
Interval may refer to:
Mathematics and physics
* Interval (mathematics), a range of numbers
** Partially ordered set#Intervals, its generalization from numbers to arbitrary partially ordered sets
* A statistical level of measurement
* Interval e ...
.
In the construction by transfinite induction, it can be shown that, in each step, the
number
A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers c ...
of sets is, at most, the
cardinality of the continuum
In set theory, the cardinality of the continuum is the cardinality or "size" of the set of real numbers \mathbb R, sometimes called the continuum. It is an infinite cardinal number and is denoted by \mathfrak c (lowercase fraktur "c") or , \mathb ...
. So, the total number of Borel sets is less than or equal to
In fact, the cardinality of the collection of Borel sets is equal to that of the continuum (compare to the number of
Lebesgue measurable
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wit ...
sets that exist, which is strictly larger and equal to
).
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-algebra of Borel sets of ''X''.
George Mackey
George Whitelaw Mackey (February 1, 1916 – March 15, 2006) was an American mathematician known for his contributions to quantum logic, representation theory, and noncommutative geometry.
Career
Mackey earned his bachelor of arts at Rice Univer ...
defined a Borel space somewhat differently, writing that it is "a set together with a distinguished σ-field of subsets called its Borel sets." 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-algebra generated by ''open'' sets (of a topological space), whereas Mackey's definition refers to a set equipped with an ''arbitrary'' σ-algebra. There exist measurable spaces that are not Borel spaces, for any choice of topology on the underlying space.
Measurable spaces form a
category
Category, plural categories, may refer to:
Philosophy and general uses
* Categorization, categories in cognitive science, information science and generally
*Category of being
* ''Categories'' (Aristotle)
*Category (Kant)
*Categories (Peirce)
* ...
in which the
morphism
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
s are
measurable function
In mathematics and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in di ...
s between measurable spaces. A function
is
measurable
In mathematics, the concept of a measure is a generalization and formalization of Geometry#Length, area, and volume, geometrical measures (length, area, volume) and other common notions, such as mass and probability of events. These seemingly ...
if it
pulls back measurable sets, i.e., for all measurable sets ''B'' in ''Y'', the set
is measurable in ''X''.
Theorem. Let ''X'' be a
Polish space
In the mathematical discipline of general topology, 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 bec ...
, that is, a topological space such that there is a
metric
Metric or metrical may refer to:
* Metric system, an internationally adopted decimal system of measurement
* An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement
Mathematics
In mathem ...
''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
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
to one of
# R,
# Z,
# 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 R
n are isomorphic.
A
standard Borel space is the Borel space associated to a
Polish space
In the mathematical discipline of general topology, 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 bec ...
. A standard Borel space is characterized up to isomorphism by its cardinality, 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
In the mathematical field of descriptive set theory, a subset of a Polish space X 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 d ...
.
Every
probability measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as ''countable additivity''. The difference between a probability measure and the more gener ...
on a standard Borel space turns it into a
standard probability space
In probability theory, a standard probability space, also called Lebesgue–Rokhlin probability space or just Lebesgue space (the latter term is ambiguous) is a probability space satisfying certain assumptions introduced by Vladimir Rokhlin i ...
.
Non-Borel sets
An example of a subset of the reals that is non-Borel, due to
Lusin, is described below. In contrast, an example of a
non-measurable set
In mathematics, a non-measurable set is a set which cannot be assigned a meaningful "volume". The mathematical existence of such sets is construed to provide information about the notions of length, area and volume in formal set theory. In Zerm ...
cannot be exhibited, though its existence can be proved.
Every
irrational number
In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integ ...
has a unique representation by an infinite
continued fraction
In mathematics, a continued fraction is an expression (mathematics), expression obtained through an iterative process of representing a number as the sum of its integer part and the multiplicative inverse, reciprocal of another number, then writ ...
:
where
is some
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
and all the other numbers
are ''positive'' integers. Let
be the set of all irrational numbers that correspond to sequences
with the following property: there exists an infinite
subsequence
In mathematics, a subsequence of a given sequence is a sequence that can be derived from the given sequence by deleting some or no elements without changing the order of the remaining elements. For example, the sequence \langle A,B,D \rangle is a ...
such that each element is a
divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
of the next element. This set
is not Borel. In fact, it is
analytic, and complete in the class of analytic sets. For more details see
descriptive set theory
In mathematical logic, descriptive set theory (DST) is the study of certain classes of "well-behaved" subsets of the real line and other Polish spaces. As well as being one of the primary areas of research in set theory, it has applications to ot ...
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.
It's important to note, that while
can be constructed in ZF, it cannot be proven to be non-Borel in ZF alone. In fact, it is consistent with ZF that
is a countable union of countable sets, so that any subset of
is a Borel set.
Another non-Borel set is an inverse image