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...ures always assign the value&nbsp;1 to the underlying set, sub-probability measures assign a value lesser than or equal to&nbsp;1 to the underlying set. Let <math> \mu </math> be a [[Measure (mathematics)|measure]] on the [[measurable space]] <math> (X, \mathcal A) </math>. ...

...ory]], '''τ-additivity''' is a certain property of [[Measure (mathematics)|measure]]s on [[topological space]]s. ...is in <math>\Sigma,</math> the measure of the union is the [[supremum]] of measures of elements of <math>\mathcal{G};</math> that is,: ...

In [[measure theory]], the '''Euler measure''' of a [[polyhedral set]] equals the [[Euler integral]] of its [[indicator ==The magnitude of an Euler measure== ...

...tensity''' of a [[Measure (mathematics)|measure]] is the average value the measure assigns to an interval of length one. Let <math> \mu </math> be a measure on the real numbers. Then the intensity <math> \overline \mu </math> of <ma ...

...e radius. Thus, if (''X'',&nbsp;''d'') is a metric space, a Borel regular measure ''&mu;'' on ''X'' is said to be '''uniformly distributed''' if ...ry rigid objects. On any "decent" metric space, the uniformly distributed measures form a one-parameter linearly dependent family: ...

...uced the concept of polynomials biorthogonal with respect to a sequence of measures, while Szegő introduced the concept of two sequences of polynomials that ar ==Polynomials biorthogonal with respect to a sequence of measures== ...

...can be used, for example, to determine whether a translate of a [[Gaussian measure]] <math>\mu</math> is equivalent to <math>\mu</math> (only when the transla ...= \bigotimes_{n \in \mathbb{N}} \nu_n</math> be the corresponding product measures on <math>\mathbb{R}^\infty</math>. Suppose also that, for each <math>n \in ...

{{Short description|Mathematical measure invariant under linear isometries}} ...a measure to [[subset]]s of ''n''-dimensional Euclidean space: [[Lebesgue measure]]. ...

...th respect to ''&mu;'' is "as big as it can be". The theory of maximising measures is relatively young and quite little is known about their general structure ...[[Borel measure|Borel]] probability measures on ''X'' that are [[invariant measure|invariant]] under ''T'', i.e., for every Borel-measurable subset ''A'' of ' ...

...]]s. The theorem makes a statement about when one can extend a probability measure to a larger [[σ-algebra]]. It is of particular interest for infinite dimens ...|first2=Jürgen |last2=Lehn |title=Two principles for extending probability measures |journal=Manuscripta Math. |number=21 |pages=43–50 |date=1977|volume=21 |do ...

{{Short description|Measure derived from a random measure}} ...d as a random measure, is for example uniquely determined by its intensity measure. <ref name="Klenke528" /> ...

In [[mathematics]], a '''''G''-measure''' is a measure <math>\mu</math> that can be represented as the weak-∗ limit of a sequence ...>-1 < r < 1, m \in \mathbb N</math>. The weak-∗ limit of this product is a measure on the circle <math>\mathbb T</math>, in the sense that for <math> f \in C( ...

...tician]] [[Christer Borell]] was a pioneer of the detailed study of convex measures on [[locally convex space]]s in the 1970s.<ref name="Borell1974">{{cite jou | title = Convex measures on locally convex spaces ...

...ch of [[mathematics]], a '''continuity set''' of a [[measure (mathematics)|measure]] {{mvar|μ}} is any [[Borel set]] {{mvar|B}} such that ...topological) [[boundary (topology)|boundary]] of {{mvar|B}}. For [[signed measure]]s, one instead asks that ...

..., most famously the [[Haar measure]], and the study of [[stationary random measure]]s. ...r=2017 |title=Random Measures, Theory and Applications|series=Probability Theory and Stochastic Modelling |volume=77 |location= Switzerland |publisher=Sprin ...

{{Short description|Theory in probability theory}} ...ace]] <math>X</math> are either [[equivalent measures]] or else [[singular measure|mutually singular]]:<ref name="Bogachev">{{cite book ...

...sup>. Spherical measure is often normalized so that it is a [[probability measure]] on the sphere, i.e. so that ''&sigma;''^''n''('''S'''<sup>''n'' ==Definition of spherical measure== ...

...''ν-transform''' is an operation that transforms a [[measure (mathematics)|measure]] or a [[point process]] into a different point process. Intuitively the ν- === For measures === ...

...'mixed binomial process''' is a special [[point process]] in [[probability theory]]. They naturally arise from restrictions of ([[mixed Poisson process|mixed ...y theory)|independent]] and let <math> \delta_x </math> denote the [[Dirac measure]] on the point <math> x </math>. ...

{{Short description|Equivalence relation on mathematical measures}} ...ity measure]]s are "the same" from the point of view of [[large deviations theory]]. ...