Borel set on arvau

Wikipedia / Borel set

In mathematics, the Borel sets of a topological space are a particular class of "well-behaved" subsets of that space. For example, whereas an arbitrary subset of the real numbers might fail to be Lebesgue measurable, every Borel set of reals is universally measurable. Which sets are Borel can be specified in a number of equivalent ways. Borel sets are named after Émile Borel.

The most usual definition goes through the notion of a σ-algebra, and is closed under countable union and complement.

Then we can define the Borel -algebra to be the smallest . is then simply an element of this -algebra.

Borel sets are important in measure theory, 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. Borel sets and the associated Borel hierarchy also play a fundamental role in descriptive set theory.

In some contexts, Borel sets are defined to be generated by the compact sets of the topological space, rather than the open sets. The two definitions are equivalent for many well-behaved spaces, including all Hausdorff σ-compact spaces, but can be different in more pathological spaces.

Generating the Borel algebra

is a metric space, the Borel algebra in the first sense may be described generatively as follows.

(that is, for any subset of the power set), let

Now define by transfinite induction is an ordinal number, in the following manner:

.
is not a limit ordinal

is the first uncountable ordinal number. That is, the Borel algebra can be generated to the first uncountable ordinal.

is closed under countable unions.

, the first uncountable ordinal.

The resulting sequence of sets is termed the Borel hierarchy.

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.

that 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 cardinality of the continuum

In fact, the cardinality of the collection of Borel sets is equal to that of the continuum (compare to the number of Lebesgue measurable).

Standard Borel spaces and Kuratowski theorems

be a topological space. The Borel space.

George Mackey 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 in which the morphisms are measurable functions is measurable if it pulls back.

Theorem be a Polish space, that is, a topological space such that there is a metric a complete separable metric space. Then as a Borel space is isomorphic to one of

,
,
a finite space.

(This result is reminiscent of Maharam's theorem.)

Considered as Borel spaces, the real line , the union of with a countable set, and are isomorphic.

A standard Borel space is the Borel space associated to a Polish space. 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.

Every probability measure on a standard Borel space turns it into a standard probability space.

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 cannot be exhibited, although the existence of such a set is implied, for example, by the axiom of choice.

Every irrational number has a unique representation by an infinite simple continued fraction

is some integer are positive with the following property: there exists an infinite subsequence such that each element is a divisor is not Borel. However, it is analytic (all Borel sets are also analytic), and complete in the class of analytic sets.

It's important to note, that while Zermelo–Fraenkel axioms is a countable union of countable sets, is a Borel set.

of an infinite parity function. However, this is a proof of existence (via the axiom of choice), not an explicit example.

Alternative non-equivalent definitions

According to Paul Halmos, a subset of a locally compact Hausdorff topological space is called a Borel set if it belongs to the smallest σ-ring containing all compact sets.

Norberg and Vervaat-algebra generated by its open subsets and its compact saturated subsets is second countable is Hausdorff).

Notes

References

External links

Formal definition of Borel Sets in the Mizar system, and the list of theorems that have been formally proved about it.