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...le]] has [[Exponential function|exponential]] tails. The result was proved in 1970 by [[Xavier Fernique]]. ...nctional]] ''ℓ'' : ''X'' → '''R''', the [[push-forward measure]] ''ℓ''_∗''μ'' defined on the [[Borel sets]] of '''R''' by ...

...> is equivalent to <math>\mu</math> (only when the translation vector lies in the [[Cameron–Martin theorem|Cameron–Martin space]] of <math>\mu</math>), o ...s on <math>\mathbb{R}^\infty</math>. Suppose also that, for each <math>n \in \mathbb{N}</math>, <math>\mu_n</math> and <math>\nu_n</math> are equivalent ...

...]]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 .../ref> The general case was shown by [[Albert Ascherl]] and [[Jürgen Lehn]] in 1977.<ref name="al">{{cite journal |first1=Albert |last1=Ascherl |first2=Jü ...

{{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 ...

...s theorem on differentiation''', named after [[Guido Fubini]], is a result in [[real analysis]] concerning the [[Derivative|differentiation]] of series o ...l <math>x \in I,</math> then for [[almost everywhere|almost any]] <math>x \in I,</math> the derivatives exist and are related as:<ref name="a"/> ...

In [[mathematical analysis]], '''Krein's condition''' provides a necessary and \quad a_k \in \mathbb{C}, \, \lambda_k \geq 0 \right\},</math> ...

...{cite book|first=Alexander S.|last=Kechris|title=Classical descriptive set theory|url=https://archive.org/details/classicaldescrip0000kech|url-access=registr ...st3=José |date=2010 |title=Measurability and Selections of Multi-Functions in Banach Spaces ...

...a probabilistic and an analytical version for [[Finite measure|finite]] [[measure space]]s. The theorem was proven in 1967 by [[János Komlós (mathematician)|János Komlós]].<ref>{{cite journal|a ...

...olutely continuous distributions with respect to the [[Lebesgue measure]] (in other words, one of the players is forbidden to use a [[pure strategy]]). ...phi_k(x)</math> are continuous functions. For <math> \mu \in M_X, \lambda \in M_Y</math>, define ...

...Arthur Rosenthal]] and has been widely applied, particularly in [[operator theory]]. ...tes that if ''K'' is a compact subset of the complex plane with [[Lebesgue measure]] zero, then any continuous complex-valued function on ''K'' can be uniform ...

...in many important cases (see Examples), and (c) the former theorem is used in the proof of the latter theorem. ...th>h : X \to Y </math> out of <math>f</math> and <math>g</math> exactly as in the [[Cantor–Bernstein–Schroeder theorem|proof of the Cantor–Bernstein–Schr ...

{{Short description|Result in measure theory}} ...tes that, if <math>f_n</math> is a sequence of integrable functions on a [[measure space]] <math>(X,\Sigma,\mu)</math> that [[Pointwise convergence#Almost eve ...

{{Short description|Mathematical construction in topology}} ...atowski theorems|Borel space]] associated with a [[Polish space]]. Except in the case of [[discrete space|discrete]] Polish spaces, the standard Borel s ...

...0019811.15510.27|bibcode=2004JSP...115..185L|arxiv=math-ph/0210027}}</ref> In 2015, [[Alexandre Eremenko]] gave a simplified proof of Stahl's theorem.<re In 2023, [[Otte Heinävaara]] proved a structure theorem for [[Hermitian matric ...

{{Short description|Area formula from geometric measure theory}} ...[geometric measure theory]] the '''area formula''' relates the [[Hausdorff measure]] of the image of a [[Lipschitz map]], while accounting for multiplicity, t ...

In [[mathematical analysis]], the '''Alexandrov theorem''', named after [[Alek In this context, having a second derivative at a point means having a second-o ...

...te book|title=Fixed Point Theorems with Applications to Economics and Game Theory|last=Border|first=Kim C.|publisher=Cambridge University Press|year=1989|isb : <math>\forall x \in X: \,\,\, f(x) \in F(x) \,.</math> ...

In the theory of [[fair cake-cutting]], the '''individual-pieces set (IPS)''' is a geometric object that represents all possible utility vectors in cake partitions. ...

{{Short description|Theorem in probability theory}} ...theory]]. It is well known that if each of two [[independence (probability theory)|independent]] [[random variables]] ξ₁ and ξ₂ has a [ ...

...027895|last=Erdős|first=P.|last2=Turán|first2=P.|title=On a problem in the theory of uniform distribution. II.|journal= Proceedings of the Koninklijke Nederl Let ''μ'' be a probability measure on the [[unit circle]] '''R'''/'''Z'''. The Erdős–Turán inequality states t ...