Uniform integrability

Uniform integrability   (Redirected from Dunford–Pettis theorem) Jump to navigation Jump to search In mathematics, uniform integrability is an important concept in real analysis, functional analysis and measure theory, and plays a vital role in the theory of martingales.

Contents 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 Citations 8 References Measure-theoretic definition Uniform integrability is an extension to the notion of a family of functions being dominated in {displaystyle 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: Let {displaystyle (X,{mathfrak {M}},mu )} be a positive measure space. A set {displaystyle Phi subset L^{1}(mu )} is called uniformly integrable if {displaystyle sup _{fin Phi }|f|_{L_{1}(mu )}0} there corresponds a {displaystyle delta >0} such that {displaystyle int _{E}|f|,dmu g}}|f|,dmu =0} where {displaystyle L_{+}^{1}(mu )={gin L^{1}(mu ):ggeq 0}} .

For finite measure spaces the following result[4] follows from Definition H: Theorem 1: If {displaystyle (X,{mathfrak {M}},mu )} is a (positive) finite measure space, then a set {displaystyle Phi subset L^{1}(mu )} 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 {displaystyle (X,{mathfrak {M}},mu )} is {displaystyle sigma } -finite, Definition H yields the following equivalency: Theorem 2: Let {displaystyle (X,{mathfrak {M}},mu )} be a {displaystyle sigma } -finite measure space, and {displaystyle hin L^{1}(mu )} be such that {displaystyle h>0} almost surely. A set {displaystyle Phi subset L^{1}(mu )} is called uniformly integrable if and only if {displaystyle sup _{fin Phi }|f|_{L_{1}(mu )}0} , there exits {displaystyle delta >0} such that {displaystyle sup _{fin Phi }int _{A}|f|,dmu 0} there exists {displaystyle delta >0} such that, for every measurable {displaystyle A} such that {displaystyle P(A)leq delta } and every {displaystyle X} in {displaystyle {mathcal {C}}} , {displaystyle operatorname {E} (|X|I_{A})leq varepsilon } .

or alternatively 2. A class {displaystyle {mathcal {C}}} of random variables is called uniformly integrable (UI) if there exists {displaystyle Kin [0,infty )} such that {displaystyle operatorname {E} (|X|I_{|X|geq K})leq varepsilon {text{ for all X}}in {mathcal {C}}} , where {displaystyle I_{|X|geq K}} is the indicator function {displaystyle I_{|X|geq K}={begin{cases}1&{text{if }}|X|geq K,\0&{text{if }}|X|0} , there exists {displaystyle a>0} such that {displaystyle P(|X|>a)leq delta } for all {displaystyle Xin {mathcal {C}}} .[8] This however, does not mean that the family of measures {displaystyle {mathcal {V}}_{mathcal {C}}:={Big {}mu _{X}:Amapsto int _{A}|X|,dP,,Xin {mathcal {C}}{Big }}} 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: Suppose {displaystyle (Omega ,{mathcal {F}},P)} is a probability space. A classed {displaystyle {mathcal {C}}} of random variables is uniformly absolutely continuous with respect to {displaystyle P} if for any {displaystyle varepsilon >0} , there is {displaystyle delta >0} such that {displaystyle E[|X|I_{A}]K)+operatorname {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 {displaystyle L^{1}(mu )} .

Dunford–Pettis theorem[13][14] A class of random variables {displaystyle X_{n}subset L^{1}(mu )} is uniformly integrable if and only if it is relatively compact for the weak topology {displaystyle sigma (L^{1},L^{infty })} . de la Vallée-Poussin theorem[15][16] The family {displaystyle {X_{alpha }}_{alpha in mathrm {A} }subset L^{1}(mu )} is uniformly integrable if and only if there exists a non-negative increasing convex function {displaystyle G(t)} such that {displaystyle lim _{tto infty }{frac {G(t)}{t}}=infty {text{ and }}sup _{alpha }operatorname {E} (G(|X_{alpha }|))

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