This is a generalization of the Heine – Borel theorem, which states that any closed and bounded subspace S of R < sup > n </ sup > is compact and therefore complete.

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

for every Borel subset U of R. Given a mixed state S, we introduce the distribution of A under S as follows:

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.

uniquely determines A and conversely, is uniquely determined by A. E < sub > A </ sub > is a boolean homomorphism from the Borel subsets of R into the lattice Q of self-adjoint projections of H. In analogy with probability theory, given a state S, we introduce the distribution of A under S which is the probability measure defined on the Borel subsets of R by

Let T be a self-adjoint operator on H, h a real-valued Borel function on R. There is a unique operator S such that

Let S be a locally compact second countable Hausdorff space equipped with its Borel σ-algebra B ( S ).

Some authors require in addition that μ ( C ) < ∞ for every compact set C. If a Borel measure μ is both inner regular and outer regular, it is called a regular Borel measure.

Note that a locally finite Borel measure automatically satisfies μ ( C ) < ∞ for every compact set C.

For any positive linear functional ψ on C < sub > c </ sub >( X ), there is a unique regular Borel measure μ on X such that

For any continuous linear functional ψ on C < sub > 0 </ sub >( X ), there is a unique regular countably additive complex Borel measure μ on X such that

* 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 compactness of U ( n ) follows from the Heine – Borel theorem and the fact that it is a closed and bounded subset of M < sub > n </ sub >( C ).

K. Chandrasekharan, Hermann Weyl, 1885 – 1985, Centenary lectures delivered by C. N. Yang, R. Penrose, A. Borel, at the ETH Zürich Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo – 1986, published for the Eidgenössische Technische Hochschule, Zürich.

Let G be a semisimple Lie group or algebraic group over, and fix a maximal torus T along with a Borel subgroup B which contains T. Let λ be an integral weight of T ; λ defines in a natural way a one-dimensional representation C < sub > λ </ sub > of B, by pulling back the representation on T = B / U, where U is the unipotent radical of B.

Another essential ingredient in Mackey's work was the assignment of a Borel structure to the dual object of a locally compact group ( specifically a locally compact separable metric group ) G. One of Mackey's important conjectures, which was eventually solved by work of James Glimm on C *- algebras was that G is type I ( meaning that all its factor representations are of type I ) if and only if the Borel structure of its dual is a standard Borel space.

A state on the composite system is a positive element ρ of the dual of C ( X × Y ), which by the Riesz-Markov theorem corresponds to a regular Borel measure on X × Y.

Given an operator T, the range of the continuous functional calculus h → h ( T ) is the ( abelian ) C *- algebra C ( T ) generated by T. The Borel functional calculus has a larger range, that is the closure of C ( T ) in the weak operator topology, a ( still abelian ) von Neumann algebra.

Borel was previously identified as Le Monnier C before being named by the IAU.

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

** Every infinite game in which is a Borel subset of Baire space is determined.

* André Weil, by A. Borel, Bull. AMS 46 ( 2009 ), 661-666.

It appeared in print in a paper written by Armand Borel with Serre.

In mathematics, specifically in measure theory, a Borel measure is defined as follows: let X be a locally compact Hausdorff space, and let be the smallest σ-algebra that contains the open sets of X ; this is known as the σ-algebra of Borel sets.

Any measure μ defined on the σ-algebra of Borel sets is called a Borel measure.

The real line R with its usual topology is a locally compact Hausdorff space, hence we can define a Borel measure on it.

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 ).

Schneider and N. Andruskiewitsch finished their long-term classification effort of pointed Hopf algebras with coradical an abelian group ( excluding primes 2, 3, 5, 7 ), especially as the above finite quotients of Just like ordinary Semisimple Lie algebra they decompose into E ´ s ( Borel part ), dual F ´ s and K ´ s ( Cartan algebra ):

The measure m is called inner regular or tight if m ( B ) is the supremum of m ( K ) for K a compact set contained in the Borel set B.

Borel used this to calculate K < sub > i </ sub >( A ) and K < sub > i </ sub >( F ) modulo torsion.

Sometimes the theorem is also referred to as the Lie – Kolchin triangularization theorem because by induction it implies that with respect to a suitable basis of V the image has a triangular shape ; in other words, the image group is conjugate in GL ( n, K ) ( where n = dim V ) to a subgroup of the group T of upper triangular matrices, the standard Borel subgroup of GL ( n, K ): the image is simultaneously triangularizable.

The theorem can be stated either for a complex semisimple Lie group G or for its compact form K. Let G be a connected complex semisimple Lie group, B a Borel subgroup of G, and X = G / B the flag variety.

His seminar in Paris in the years after 1945 covered ground on several complex variables, sheaf theory, spectral sequences and homological algebra, in a way that deeply influenced Jean-Pierre Serre, Armand Borel, Alexander Grothendieck and Frank Adams, amongst others of the leading lights of the younger generation.

Instead Armand Borel and Jean-Pierre Serre wrote up and published Grothendieck's preliminary ( as he saw it ) proof.

Serre and Armand Borel subsequently organized a seminar at Princeton to understand it.

The final published paper was in effect the Borel – Serre exposition.

The interpolation formula, as it is commonly called, dates back to the works of E. Borel in 1898, and E. T. Whittaker in 1915, and was cited from works of J. M. Whittaker in 1935, and in the formulation of the Nyquist – Shannon sampling theorem by Claude Shannon in 1949.

* Prix Goncourt: J. Borel, L ' Adoration

* A. Borel, J. Tits, Groupes réductifs Publ.

* Théodore Borel, Life of Count Agénor de Gasparin ( New York, G. P. Putnam's Sons, 1881 )

In this case, the probability P ( Y = y ) = 0, and the Borel – Kolmogorov paradox demonstrates the ambiguity of attempting to define conditional probability along these lines.

In general, when the Hilbert space is not finite-dimensional, the normal operator N gives rise to a projection-valued measure P on its spectrum, σ ( N ), which assigns a projection P < sub > Ω </ sub > to each Borel subset of σ ( N ).

Moreover G has an abelian subgroup Q of order prime to p containing an element y such that P < sub > 0 </ sub > normalizes Q and ( P < sub > 0 </ sub >)< sup > y </ sup > normalizes U, where P < sub > 0 </ sub > is the additive group of the finite field of order p. ( For p = 2 a similar configuration occurs in the group SL < sub > 2 </ sub >( 2 < sup > q </ sup >), with PU a Borel subgroup of upper trianguar matrices and Q the subgroup of order 3 generated by y =().