specifically 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...
10 KB (1,239 words) - 17:28, 29 January 2024
an outer measure μ on n-dimensional Euclidean space Rn is called a Borel regular measure if the following two conditions hold: Every Borel set B ⊆ Rn...
2 KB (247 words) - 05:12, 23 December 2021
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...
13 KB (1,793 words) - 02:48, 4 June 2024
mathematician, he was known for his founding work in the areas of measure theory and probability. Borel was born in Saint-Affrique, Aveyron, the son of a Protestant...
13 KB (1,208 words) - 07:15, 6 May 2024
mathematics (specifically in measure theory), a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological...
19 KB (2,697 words) - 10:22, 11 May 2024
Lebesgue-measurable sets than there are Borel measurable sets. The Borel measure is translation-invariant, but not complete. The Haar measure can be defined on any locally...
18 KB (2,641 words) - 20:13, 3 May 2024
set that is not contained in the Borel sets. Hence, the Borel measure is not complete. n-dimensional Lebesgue measure is the completion of the n-fold product...
6 KB (826 words) - 17:41, 19 March 2024
theory, the Borel–Cantelli lemma is a theorem about sequences of events. In general, it is a result in measure theory. It is named after Émile Borel and Francesco...
11 KB (1,815 words) - 21:26, 10 May 2024
s\in S\}.} Left and right translates map Borel sets onto Borel sets. A measure μ {\displaystyle \mu } on the Borel subsets of G {\displaystyle G} is called...
31 KB (5,360 words) - 23:32, 29 December 2023
other Borel sets is a Borel probability measure that is neither inner regular nor outer regular. Borel regular measure Radon measure Regularity theorem for...
6 KB (863 words) - 19:39, 18 January 2024
In mathematics, Gaussian measure is a Borel measure on finite-dimensional Euclidean space R n {\displaystyle R^{n}} , closely related to the normal distribution...
6 KB (1,005 words) - 00:04, 11 May 2024
Lebesgue–Stieltjes integration (redirect from Lebesgue-Stieltjes measure)
Lebesgue–Stieltjes measure, which may be associated to any function of bounded variation on the real line. The Lebesgue–Stieltjes measure is a regular Borel measure, and...
11 KB (1,624 words) - 06:45, 6 February 2024
spectrum) of a measure μ {\displaystyle \mu } on a measurable topological space ( X , Borel ( X ) ) {\displaystyle (X,\operatorname {Borel} (X))} is a...
11 KB (1,918 words) - 09:58, 21 July 2023
operators, in which case the PVM is sometimes referred to as the spectral measure. The Borel functional calculus for self-adjoint operators is constructed using...
16 KB (2,492 words) - 08:06, 31 May 2024
measure spaces, the product space may not be. Consequently, the completion procedure is needed to extend the Borel measure into the Lebesgue measure,...
5 KB (928 words) - 09:26, 19 April 2024
Measurable function (redirect from Borel function)
{\displaystyle (Y,T)} are Borel spaces, a measurable function f : ( X , Σ ) → ( Y , T ) {\displaystyle f:(X,\Sigma )\to (Y,T)} is also called a Borel function. Continuous...
9 KB (1,329 words) - 00:07, 7 December 2023
support, and the measures can be Baire measures or regular Borel measures or Radon measures or signed measures or complex measures. The statement of...
9 KB (1,119 words) - 08:41, 15 May 2024
centuries that measure theory became a branch of mathematics. The foundations of modern measure theory were laid in the works of Émile Borel, Henri Lebesgue...
34 KB (5,471 words) - 13:55, 6 June 2024
chemist Borel (crater), a lunar crater, named after Émile Borel Borel algebra, operating on Borel sets, named after Émile Borel, also: Borel measure, the...
1 KB (174 words) - 14:05, 17 May 2024
Σ-algebra (redirect from Probability measure space)
(a construction known as the Borel hierarchy). There are at least three key motivators for σ-algebras: defining measures, manipulating limits of sets...
30 KB (5,337 words) - 13:11, 26 March 2024
Null set (redirect from Measure zero)
have measure zero. Lebesgue measure is an example of a complete measure; in some constructions, it is defined as the completion of a non-complete Borel measure...
11 KB (1,730 words) - 10:42, 12 February 2024
least as fine as the Borel σ {\displaystyle \sigma } -algebra on X {\displaystyle X} ). A measure/signed measure/complex measure μ {\displaystyle \mu...
2 KB (348 words) - 20:55, 28 December 2023
usual Lebesgue measure cannot be extended to all infinite-dimensional spaces. This is because any translation-invariant Borel measure on an infinite-dimensional...
6 KB (945 words) - 18:43, 13 June 2024
principal value Measure (mathematics) Sigma algebra Separable sigma algebra Filtration (abstract algebra) Borel algebra Borel measure Indicator function...
2 KB (221 words) - 02:51, 2 May 2022
of these measures, and their convolution in particular. Borel measure – Measure defined on all open sets of a topological space Fuzzy measure – theory...
7 KB (970 words) - 22:35, 31 March 2024
specifically, in geometric measure theory — spherical measure σn is the "natural" Borel measure on the n-sphere Sn. Spherical measure is often normalized so...
5 KB (698 words) - 05:38, 23 December 2021
true for tempered distributions. The Fourier transform of a finite Borel measure μ on Rn is given by: μ ^ ( ξ ) = ∫ R n e − i 2 π x ⋅ ξ d μ . {\displaystyle...
177 KB (21,038 words) - 21:59, 30 May 2024
Convolution (section Measures)
(Hörmander 1983, §4.2). The convolution of any two Borel measures μ and ν of bounded variation is the measure μ ∗ ν {\displaystyle \mu *\nu } defined by (Rudin...
66 KB (8,694 words) - 20:56, 31 May 2024
defined using the push-forward and the standard Gaussian measure on the real line: a Borel measure γ on a separable Banach space X is called Gaussian if...
6 KB (959 words) - 01:13, 6 May 2024
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...
3 KB (423 words) - 07:18, 27 May 2024