The deeper structure theory applies to connected linear algebraic groups G, and begins with the definition of Borel subgroups B.

The most important subgroups of a linear algebraic group, besides its Borel subgroups, are its tori, especially the maximal ones ( similar to the study of maximal tori in Lie groups ).

there is a single conjugacy class of Borel subgroups.

Subgroups between a Borel subgroup B and the ambient group G are called parabolic subgroups.

Working over algebraically closed fields, the Borel subgroups turn out to be the minimal parabolic subgroups in this sense.

For a simple algebraic group G, the set of conjugacy classes of parabolic subgroups is in bijection with the set of all subsets of nodes of the corresponding Dynkin diagram ; the Borel subgroup corresponds to the empty set and G itself corresponding to the set of all nodes.

The subgroups of G containing conjugates of B are the parabolic subgroups ; conjugates of B are called Borel subgroups ( or minimal parabolic subgroups ).

In this case, is the smallest σ-algebra that contains the open intervals of R. While there are many Borel measures μ, the choice of Borel measure which assigns for every interval is sometimes called " the " Borel measure on R. In practice, even " the " Borel measure is not the most useful measure defined on the σ-algebra of Borel sets ; indeed, the Lebesgue measure is an extension of " the " Borel measure which possesses the crucial property that it is a complete measure ( unlike the Borel measure ).

Classical measure theory makes deep usage of the axiom of choice, which is fundamental to, first, distinction between measurable and non-measurable sets, the existence of the latter being behind such famous results as the Banach – Tarski paradox, and secondly the hierarchies of notions of measure captured by notions such as Borel algebras, which are an important source of intuitions in set theory.

A more useful example is the set of subsets of the real line formed by starting with all open intervals and adding in all countable unions, countable intersections, and relative complements and continuing this process until the relevant closure properties are achieved ( a construction known as the Borel σ-algebra Borel set ).

For example, many results about Borel sets are proved by transfinite induction on the ordinal rank of the set ; these ranks are already well-ordered, so the axiom of choice is not needed to well-order them.

Borel sets are named after Émile Borel.

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.

In some contexts, Borel sets are defined to be generated by the compact sets of the topological space, rather than the open sets.

* If ( X, Σ ) and ( Y, Τ ) are Borel spaces, a measurable function f: ( X, Σ ) → ( Y, Τ ) is also called a Borel function.

Continuous functions are Borel functions but not all Borel functions are continuous.

* A Lebesgue measurable function is a measurable function, where is the sigma algebra of Lebesgue measurable sets, and is the Borel algebra on the complex numbers C. Lebesgue measurable functions are of interest in mathematical analysis because they can be integrated.

* The Euclidean spaces R < sup >< var > n </ var ></ sup > ( and in particular the real line R ) are locally compact as a consequence of the Heine – Borel theorem.

To clarify, when one says that the Lebesgue measure is an extension of the Borel measure, it means that every Borel measurable set E is also a Lebesgue measurable set, and the Borel measure and the Lebesgue measure coincide on the Borel sets ( i. e., for every Borel measurable set ).

In general, one uses the Borel functional calculus to calculate.

A measure μ on the Borel subsets of G is called left-translation-invariant if for all Borel subsets S of G and all a in G one has

Émile Borel was one of the first mathematicians to formally address randomness in 1909.

One of the most remarkable facts about a locally compact group G is that it carries an essentially unique natural measure, the Haar measure, which allows one to consistently measure the " size " of sufficiently regular subsets of G. " Sufficiently regular subset " here means a Borel set ; that is, an element of the σ-algebra generated by the compact sets.

For the purposes of functional analysis, and in particular of harmonic analysis, one wishes to carry over the group ring construction to topological groups G. In case G is a locally compact Hausdorff group, G carries an essentially unique left-invariant countably additive Borel measure μ called Haar measure.

However, a purely algebraic theory was first developed by, with Armand Borel as one of its pioneers.

It is possible to verify that these steps produce a theory identical with the one that starts from a Radon measure defined as a function that assigns a number to each Borel set of X.

) Luzin proved more generally that any two disjoint analytic sets are separated by a Borel set: in other words there is a Borel set containing one and disjoint from the other.

For technical reasons, one needs to consider separately the positive and negative parts of A defined by the Borel functional calculus for unbounded operators.

By the assumption that the energy eigenvalues diverge, the Hamiltonian H is an unbounded operator, therefore we have invoked the Borel functional calculus to exponentiate the Hamiltonian H. Alternatively, in non-rigorous fashion, one can consider that to be the exponential power series.

Given a countably additive measure μ on X, a measurable set is one that differs from a Borel set by a null set.

Let G be the complex special linear group SL ( 2, C ), with a Borel subgroup consisting of upper triangular matrices with determinant one.

Given Borel equivalence relations E and F on Polish spaces X and Y respectively, one says that E is Borel reducible to F, in symbols E ≤< sub > B </ sub > F, if and only if there is a Borel function

Logitech International S. A. was co-founded in Apples, Vaud, Switzerland, in 1981 by two Stanford PhD alumni, Daniel Borel and Pierluigi Zappacosta, Jean-luc Mazzone and Giacomo Marini, formerly a manager at Olivetti.

A fundamental result shows that any two uncountable Polish spaces X and Y 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.

Between any two uncountable Polish spaces, there is a Borel isomorphism ; that is, a bijection that preserves the Borel structure.

In mathematics, an outer measure μ on n-dimensional Euclidean space R < sup > n </ sup > is called Borel regular if the following two conditions hold:

An outer measure satisfying only the first of these two requirements is called a Borel measure, while an outer measure satisfying only the second requirement is called a regular measure.

There exists an example of two sets of uniqueness whose union is not a set of uniqueness, but the sets in this example are not Borel.

It is an open problem whether the union of any two Borel sets of uniqueness is a set of uniqueness.

Kuratowski's theorem then states that two standard Borel spaces X and Y are Borel-isomorphic iff | X |

A key component of the proof implicitly showed determinacy of parity games, which lie in the third level of the Borel hierarchy.