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	~~In mathematics~~, ''~~'Scheffé~~'s ~~lemma'''~~ is a ~~proposition~~ in [~~[measure theory~~]~~] concerning~~ the ~~[[Convergence (mathematics)\|convergence]]~~ of ~~sequences~~ of ~~[[integral]]~~s~~. It states~~ that, if <~~math~~>~~f_n~~<~~/math~~> is a ~~sequence~~ of ~~[[Lebesgue integration\|integrable functions]]~~ on ~~a [[measure space]]~~ <~~math~~>(X,~~\Sigma~~,~~\mu)~~<~~/math~~> that [~~[Pointwise convergence#Almost everywhere convergence\|converges almost~~ everywhere]] to ~~another integrable function~~ <~~math~~>f<~~/math~~>~~, then~~ <~~math~~>~~\int \|f_n~~ - ~~f\| \, d\mu \~~to 0<~~/math~~> ~~if and only if~~ <~~math~~>~~\int \| f_n \| \~~, ~~d\mu \to \int \| f \| \~~, ~~d\mu~~<~~/math~~>.<~~ref name=Williams~~>~~{{cite book\|author=David Williams\|title=Probability with Martingales\|publisher=Cambridge University Press\|location=New York\|year=1991\|ref=harv\|page=55}}~~</~~ref>~~		<br><br>Hostgator is a big independently had hosting business operating from numerous state of the art facilities in Dallas, Texas. 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	==~~Applications~~==
	~~Applied to [[probability theory]~~], ~~Scheffe~~'s ~~theorem states~~ that ~~almost everywhere pointwise convergence~~ of the ~~[[probability density function]]s of~~ a ~~sequence of [[random variable]]s implies [[convergence in distribution]]~~.

	~~==History==~~
	~~[[Henry Scheffé]] published~~ a ~~proof~~ of the ~~statement~~ on ~~convergence of probability densities in 1947~~. ~~The result however~~ is ~~a special case~~ of ~~a theorem by [[Frigyes Riesz]] about convergence in [[Lp spaces\|L~~<~~sup~~>~~''p''~~<~~/sup~~> ~~spaces]] published in 1928.~~<~~ref~~>~~{{cite journal\|journal=Periodica Mathematica Hungarica\|date=September 2010\|volume=61\|number=1-2\|pages=225–229\|title=~~
	~~Why~~ the ~~theorem~~ of ~~Scheffé should be rather called a theorem of Riesz\|author=Norbert Kusolitsch\|url=http://link~~.~~springer.com~~/~~article/10~~.~~1007/s10998-010-3225-6}}</ref>~~

	~~== References ==~~
	~~{{Reflist}}~~

	~~{{DEFAULTSORT:Scheffe's lemma}}~~
	~~[[Category:Theorems in measure theory]]~~

en>Waacstats: /* Bibliography */Add persondata short description using AWB

2014-01-01T14:26:26Z

Bibliography: Add persondata short description using AWB

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In mathematics, '''Scheffé's lemma''' is a proposition in [[measure theory]] concerning the [[Convergence (mathematics)|convergence]] of sequences of [[integral]]s. It states that, if <math>f_n</math> is a sequence of [[Lebesgue integration|integrable functions]] on a [[measure space]] <math>(X,\Sigma,\mu)</math> that [[Pointwise convergence#Almost everywhere convergence|converges almost everywhere]] to another integrable function <math>f</math>, then <math>\int |f_n - f| \, d\mu \to 0</math> if and only if <math>\int | f_n | \, d\mu \to \int | f | \, d\mu</math>.<ref name=Williams>{{cite book|author=David Williams|title=Probability with Martingales|publisher=Cambridge University Press|location=New York|year=1991|ref=harv|page=55}}</ref>

==Applications==
Applied to [[probability theory]], Scheffe's theorem states that almost everywhere pointwise convergence of the [[probability density function]]s of a sequence of [[random variable]]s implies [[convergence in distribution]].

==History==
[[Henry Scheffé]] published a proof of the statement on convergence of probability densities in 1947. The result however is a special case of a theorem by [[Frigyes Riesz]] about convergence in [[Lp spaces|L<sup>''p''</sup> spaces]] published in 1928.<ref>{{cite journal|journal=Periodica Mathematica Hungarica|date=September 2010|volume=61|number=1-2|pages=225–229|title=
Why the theorem of Scheffé should be rather called a theorem of Riesz|author=Norbert Kusolitsch|url=http://link.springer.com/article/10.1007/s10998-010-3225-6}}</ref>

== References ==
{{Reflist}}

{{DEFAULTSORT:Scheffe's lemma}}
[[Category:Theorems in measure theory]]

Alexandros Chapsiadis - Revision history

en>Ser Amantio di Nicolao: /* Bibliography */add category using AWB

en>Waacstats: /* Bibliography */Add persondata short description using AWB