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 after Émile Borel.
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, 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.
In mathematics, a Borelset is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations...
are Borel isomorphic: there is a bijection from X to Y such that the preimage of any Borelset is Borel, and the image of any Borelset is Borel. This...
in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all Borelsets). Some authors require...
algebra are called Borel sets. Each Borelset is assigned a unique countable ordinal number called the rank of the Borelset. The Borel hierarchy is of particular...
n-dimensional Euclidean space Rn is called a Borel regular measure if the following two conditions hold: Every Borelset B ⊆ Rn is μ-measurable in the sense of...
measure theory, the Baire sets form a σ-algebra of a topological space that avoids some of the pathological properties of Borelsets. There are several inequivalent...
Moreover, every Borelset is Lebesgue-measurable. However, there are Lebesgue-measurable sets which are not Borelsets. Any countable set of real numbers...
of a non-complete Borel measure. The Borel measure is not complete. One simple construction is to start with the standard Cantor set K , {\displaystyle...
of a Borelset in a Polish space. A is a Suslin set, the image of the Suslin operation. There is a Polish space Y {\displaystyle Y} and a Borelset B ⊆...
In mathematics, a standard Borel space is the Borel space associated to a Polish space. Discounting Borel spaces of discrete Polish spaces, there is, up...
assigns measure 1 to Borelsets containing an unbounded closed subset of the countable ordinals and assigns 0 to other Borelsets is a Borel probability measure...
of Borelsets of a Hausdorff topological space X that is finite on all compact sets, outer regular on all Borelsets, and inner regular on open sets. These...
In descriptive set theory, the Borel determinacy theorem states that any Gale–Stewart game whose payoff set is a Borelset is determined, meaning that...
subsets of G {\displaystyle G} is called the Borel algebra. An element of the Borel algebra is called a Borelset. If g {\displaystyle g} is an element of...
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...
after Émile BorelBorel algebra, operating on Borelsets, named after Émile Borel, also: Borel measure, the measure on a Borel algebra Borel distribution...
distribution can be generalized to more complicated sets than intervals. Let S {\displaystyle S} be a Borelset of positive, finite Lebesgue measure λ ( S )...
hierarchy. Many properties of Borelsets can be established in ZFC, but proving these properties hold for more complicated sets requires additional axioms...
{\mathfrak {B}}} is the standard Borelset on the real line) F {\displaystyle {\mathfrak {F}}} – a non-empty collection of Borelsets F ⊆ B {\displaystyle {\mathfrak...
example. This set A {\displaystyle \textstyle A} is formed by a finite number of unions, whereas a Borelset is formed by a countable number of set operations...
for any Borelset A, in which the integral is Lebesgue. the cumulative distribution function of X is absolutely continuous. for any Borelset A of real...
In measure theory, a branch of mathematics, a continuity set of a measure μ is any Borelset B such that μ ( ∂ B ) = 0 , {\displaystyle \mu (\partial...
a set that is both analytic and coanalytic is necessarily Borel), the inverse of any such measurable bijective function is also measurable. Borel isomorphisms...
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