In mathematics, 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 ...

. Discounting Borel spaces of

discrete Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory *Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit *Discrete group, a ...

Polish spaces, there is, up to

isomorphism 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 i ...

measurable space In mathematics, a measurable space or Borel space is a basic object in measure theory. It consists of a set and a σ-algebra, which defines the subsets that will be measured. Definition Consider a set X and a σ-algebra \mathcal A on X. Then ...

s, only one standard Borel space.

Formal definition

(X, \Sigma)

is said to be "standard Borel" if there exists 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 ...

X

that makes it a

complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies ...

separable metric space in such a way that

\Sigma

is then the

Borel σ-algebra 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 a ...

. Standard Borel spaces have several useful properties that do not hold for general measurable spaces.

Properties

* If

(X, \Sigma)

and

(Y, T)

are standard Borel then any bijective measurable mapping

f : (X, \Sigma) \to (Y, \Tau)

is an isomorphism (that is, the inverse mapping is also measurable). This follows from Souslin's theorem, as a set that is both analytic and coanalytic is necessarily Borel. * If

(X, \Sigma)

and

(Y, T)

are standard Borel spaces and

f : X \to Y

then

f

is measurable if and only if the

graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...

f

is Borel. * The product and direct union of a

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 number ...

family of standard Borel spaces are standard. * 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 g ...

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 ...

Kuratowski's theorem

Theorem. Let

X

be a

, that is, 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 po ...

such that there is a

d

X

that defines the topology of

X

and that makes

X

a complete separable metric space. Then

X

as a Borel space is Borel isomorphic to one of (1)

\R,

(2)

\Z

or (3) a finite discrete space. (This result is reminiscent of

Maharam's theorem In mathematics, Maharam's theorem is a deep result about the decomposability of measure spaces, which plays an important role in the theory of Banach spaces. In brief, it states that every complete measure space is decomposable into "non-atomic par ...

.) It follows that a standard Borel space is characterized up to isomorphism by its

cardinality In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...

, and that any uncountable standard Borel space has 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 , \ma ...

. Borel isomorphisms on standard Borel spaces are analogous to

homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...

s on

s: both are bijective and closed under composition, and a homeomorphism and its inverse are both

continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...

, instead of both being only Borel measurable.

References

{{Measure theory Descriptive set theory General topology Measure theory Space (mathematics)

Formal definition

Properties

Kuratowski's theorem

See also

References