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...

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...

defined on all 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...

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...

other Borel sets is a Borel probability measure that is neither inner regular nor outer regular. Borel regular measure Radon measure Regularity theorem for...

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...

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...

and its Lebesgue measure is ( b − a ) ( c − d ) {\textstyle (b-a)(c-d)} , the area of the corresponding rectangle. Moreover, every Borel set is Lebesgue-measurable...

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...

support, and the measures can be Baire measures or regular Borel measures or Radon measures or signed measures or complex measures. The statement of...

In mathematics, a Gaussian measure is a Borel measure on finite-dimensional Euclidean space R n {\displaystyle \mathbb {R} ^{n}} , closely related to the...

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...

spectrum) of a measure μ {\displaystyle \mu } on a measurable topological space ( X , Borel ⁡ ( X ) ) {\displaystyle (X,\operatorname {Borel} (X))} is a...

chemist Borel (crater), a lunar crater, named after Émile Borel Borel algebra, operating on Borel sets, named after Émile Borel, also: Borel measure, the...

principal value Measure (mathematics) Sigma algebra Separable sigma algebra Filtration (abstract algebra) Borel algebra Borel measure Indicator function...

Lebesgue measure cannot be straightforwardly extended to all infinite-dimensional spaces due to a key limitation: any translation-invariant Borel measure on...

A be a Borel subset of Rn, and let s > 0. Then the following are equivalent: Hs(A) > 0, where Hs denotes the s-dimensional Hausdorff measure. There is...

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...

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...

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...

{\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...

measure spaces, the product space may not be. Consequently, the completion procedure is needed to extend the Borel measure into the Lebesgue measure,...

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...

measure on the circle. One example of a finite Borel measure that is not a function is the Dirac measure. Its Fourier transform is a constant function...

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

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...

specifically, in geometric measure theory — spherical measure σn is the "natural" Borel measure on the n-sphere Sn. Spherical measure is often normalized so...

(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...

probability measure on X with precisely the same null sets as μ. A Borel measure (in the sense of a locally finite measure on the Borel σ {\displaystyle...

Properties of sets of small finite ranks are important in measure theory and analysis. The Borel algebra in an arbitrary topological space is the smallest...