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...

are Borel isomorphic: there is a bijection from X to Y such that the preimage of any Borel set is Borel, and the image of any Borel set is Borel. This...

algebra are called Borel sets. Each Borel set is assigned a unique countable ordinal number called the rank of the Borel set. The Borel hierarchy is of particular...

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...

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...

theorem Borel right process Borel set Borel summation Borel distribution Borel's conjecture about strong measure zero sets (not to be confused with Borel conjecture...

Moreover, every Borel set is Lebesgue-measurable. However, there are Lebesgue-measurable sets which are not Borel sets. Any countable set of real numbers...

of a Borel set 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 Borel set B ⊆...

of intervals (called Borel sets) plus-minus null sets. These sets are rich enough to include every conceivable definition of a set that arises in standard...

of Borel sets of a Hausdorff topological space X that is finite on all compact sets, outer regular on all Borel sets, and inner regular on open sets. These...

construction known as the Borel hierarchy). There are at least three key motivators for σ-algebras: defining measures, manipulating limits of sets, and managing partial...

n-dimensional Euclidean space Rn is called a Borel regular measure if the following two conditions hold: Every Borel set B ⊆ Rn is μ-measurable in the sense of...

subsets of G {\displaystyle G} is called the Borel algebra. An element of the Borel algebra is called a Borel set. If g {\displaystyle g} is an element of...

In set theory, a subset of a Polish space X {\displaystyle X} is ∞-Borel if it can be obtained by starting with the open subsets of X {\displaystyle X}...

a standard Borel space is the Borel space associated with a Polish space. Except in the case of discrete Polish spaces, the standard Borel space is unique...

assigns measure 1 to Borel sets containing an unbounded closed subset of the countable ordinals and assigns 0 to other Borel sets is a Borel probability measure...

hierarchy. Many properties of Borel sets can be established in ZFC, but proving these properties hold for more complicated sets requires additional axioms...

measure theory, the Baire sets form a σ-algebra of a topological space that avoids some of the pathological properties of Borel sets. There are several inequivalent...

paper from 1905 he wrote that the projection of Borel set in the plane onto the real line is again a Borel set. The mathematician Mikhail Yakovlevich Suslin...

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...

example. This set A {\displaystyle \textstyle A} is formed by a finite number of unions, whereas a Borel set is formed by a countable number of set operations...

In descriptive set theory, the Borel determinacy theorem states that any Gale–Stewart game whose payoff set is a Borel set is determined, meaning that...

In measure theory, a branch of mathematics, a continuity set of a measure μ is any Borel set B such that μ ( ∂ B ) = 0 , {\displaystyle \mu (\partial...

for any Borel set A, in which the integral is Lebesgue. the cumulative distribution function of X is absolutely continuous. for any Borel set A of real...

after Émile Borel Borel algebra, operating on Borel sets, named after Émile Borel, also: Borel measure, the measure on a Borel algebra Borel distribution...

a construction is called a Borel code. Given a Borel set B {\displaystyle B} in V {\displaystyle V} , one recovers a Borel code, and then applies the...

In real analysis the Heine–Borel theorem, named after Eduard Heine and Émile Borel, states: For a subset S {\displaystyle S} of Euclidean space R n {\displaystyle...

today because they are the primary setting for descriptive set theory, including the study of Borel equivalence relations. Polish spaces are also a convenient...

Descriptive set theory Analytic set Analytical hierarchy Borel equivalence relation Infinity-Borel set Lightface analytic game Perfect set property Polish...

measure 0. The Cantor set is an example of an uncountable set of Lebesgue measure 0 which is not of strong measure zero. Borel's conjecture states that...