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
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
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
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
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...
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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
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
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
measure spaces, the product space may not be. Consequently, the completion procedure is needed to extend the Borel measure into the Lebesgue measure,...
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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
In compact metric spaces the Borel sets and the Baire sets are the same, so Baire measures are the same as Borel measures that are finite on compact sets...
2 KB (304 words) - 11:17, 20 October 2023
chemist Borel (crater), a lunar crater, named after Émile Borel Borel algebra, operating on Borel sets, named after Émile Borel, also: Borel measure, the...
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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
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
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
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
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
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
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
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
Baire set (category Measure theory)
specifically in measure theory, the Baire sets form a σ-algebra of a topological space that avoids some of the pathological properties of Borel sets. There...
9 KB (1,204 words) - 01:14, 17 December 2023
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