# Uniform integrability 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: 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 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}}} . 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 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 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 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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