Uniform integrability

Uniform integrability (Redirected from Dunford–Pettis theorem) Zur Navigation springen Zur Suche springen In der Mathematik, uniform integrability is an important concept in real analysis, functional analysis and measure theory, and plays a vital role in the theory of martingales.

Inhalt 1 Measure-theoretic definition 2 Probability definition 3 Tightness and uniform integrability 4 Related corollaries 5 Relevant theorems 6 Relation to convergence of random variables 7 Zitate 8 References Measure-theoretic definition Uniform integrability is an extension to the notion of a family of functions being dominated in {Anzeigestil L_{1}} which is central in dominated convergence. Several textbooks on real analysis and measure theory often use the following definition:[1][2] Definition A: Lassen {Anzeigestil (X,{mathfrak {M}},in )} be a positive measure space. A set {displaystyle Phi subset L^{1}(in )} is called uniformly integrable if {displaystyle sup _{fin Phi }|f|_{L_{1}(in )}0} there corresponds a {displaystyle delta >0} such that {displaystyle int _{E}|f|,dmu g}}|f|,dmu = 0} wo {Anzeigestil L_{+}^{1}(in )={gin L^{1}(in ):ggeq 0}} .

For finite measure spaces the following result[4] follows from Definition H: Satz 1: Wenn {Anzeigestil (X,{mathfrak {M}},in )} ist ein (positiv) finite measure space, then a set {displaystyle Phi subset L^{1}(in )} is called uniformly integrable if and only if {displaystyle inf _{ageq 0}sup _{fin Phi }int _{{|f|>a}}|f|,dmu = 0} May textbooks in probability present Theorem 1 as the definition of uniform integrability in Probability spaces. When the space {Anzeigestil (X,{mathfrak {M}},in )} ist {Display-Sigma } -endlich, Definition H yields the following equivalency: Satz 2: Lassen {Anzeigestil (X,{mathfrak {M}},in )} sei ein {Display-Sigma } -finite measure space, und {displaystyle hin L^{1}(in )} be such that {displaystyle h>0} fast sicher. A set {displaystyle Phi subset L^{1}(in )} is called uniformly integrable if and only if {displaystyle sup _{fin Phi }|f|_{L_{1}(in )}0} , there exits {displaystyle delta >0} so dass {displaystyle sup _{fin Phi }int _{EIN}|f|,dmu 0} es existiert {displaystyle delta >0} so dass, for every measurable {Anzeigestil A} so dass {Anzeigestil P(EIN)leq delta } Und jeder {Anzeigestil X} in {Anzeigestil {mathematisch {C}}} , {Anzeigestil Betreibername {E} (|X|ICH_{EIN})leq varepsilon } .

or alternatively 2. A class {Anzeigestil {mathematisch {C}}} of random variables is called uniformly integrable (UI) if there exists {displaystyle Kin [0,unendlich )} so dass {Anzeigestil Betreibername {E} (|X|ICH_{|X|geq K})leq varepsilon {Text{ for all X}}in {mathematisch {C}}} , wo {Anzeigestil I_{|X|geq K}} is the indicator function {Anzeigestil I_{|X|geq K}={Start{Fälle}1&{Text{wenn }}|X|geq K,\0&{Text{wenn }}|X|0} , es existiert {displaystyle a>0} so dass {Anzeigestil P(|X|>a)leq delta } für alle {displaystyle Xin {mathematisch {C}}} .[8] This however, does not mean that the family of measures {Anzeigestil {mathematisch {v}}_{mathematisch {C}}:={Groß {}in _{X}:Amapsto int _{EIN}|X|,dP,,Xin {mathematisch {C}}{Groß }}} is tight.

There is another notion of uniformity, slightly different than uniform integrability, which also has many applications in Probability and measure theory, and which does not require random variables to have a finite integral[9] Definition: Vermuten {Anzeigestil (Omega ,{mathematisch {F}},P)} is a probability space. A classed {Anzeigestil {mathematisch {C}}} of random variables is uniformly absolutely continuous with respect to {Anzeigestil P} if for any {displaystyle varepsilon >0} , there is {displaystyle delta >0} so dass {Anzeigestil E[|X|ICH_{EIN}]K)+Name des Bedieners {E} (|X|,|X|1} ) is uniformly integrable. Relevant theorems In the following we use the probabilistic framework, but regardless of the finiteness of the measure, by adding the boundedness condition on the chosen subset of {Anzeigestil L^{1}(in )} .

Dunford–Pettis theorem[13][14] A class of random variables {Anzeigestil X_{n}subset L^{1}(in )} is uniformly integrable if and only if it is relatively compact for the weak topology {Display-Sigma (L^{1},L^{unendlich })} . de la Vallée-Poussin theorem[15][16] The family {Anzeigestil {X_{Alpha }}_{alpha in mathrm {EIN} }subset L^{1}(in )} is uniformly integrable if and only if there exists a non-negative increasing convex function {Anzeigestil G(t)} so dass {Anzeigestil lim _{tto infty }{frac {G(t)}{t}}=infty {Text{ und }}sup _{Alpha }Name des Bedieners {E} (G(|X_{Alpha }|))

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