{{Short description|Mathematical concept}} 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.
==Measure-theoretic definition== Uniform integrability is an extension to the notion of a family of functions being dominated in <math> L^1</math> which is central in dominated convergence. Several textbooks on real analysis and measure theory use the following definition:<ref>{{cite book|author1=Royden, H.L. |author2=Fitzpatrick, P.M. |name-list-style=amp |year=2010|title=Real Analysis|edition=4|publisher=Prentice Hall|location=Boston|page=93|isbn=978-0-13-143747-0}}</ref>
<blockquote> '''Definition A:''' Let <math> (X,\mathfrak{M}, \mu)</math> be a positive measure space. A set <math>\Phi\subset L^1(\mu)</math> is called '''uniformly integrable''' if <math>\sup_{f\in\Phi}\|f\|_{L^1(\mu)}<\infty</math>, and to each <math> \varepsilon>0 </math> there corresponds a <math> \delta>0 </math> such that
: <math> \int_E |f| \, d\mu < \varepsilon </math>
whenever <math>f \in \Phi </math> and <math>\mu(E)<\delta.</math> </blockquote>
Definition A is rather restrictive for infinite measure spaces. A more general definition<ref>{{cite book|first=G. A.|last=Hunt|year=1966|title=Martingales et Processus de Markov|publisher=Dunod|location=Paris|page=33}}</ref> of uniform integrability that works well in general measure spaces was introduced by G. A. Hunt.
<blockquote> '''Definition H:''' Let <math> (X,\mathfrak{M},\mu)</math> be a positive measure space. A set <math> \Phi\subset L^1(\mu)</math> is called '''uniformly integrable''' if and only if
:<math> \inf_{g\in L^1_+(\mu)}\sup_{f\in\Phi}\int_{\{|f|>g\}}|f|\, d\mu=0 </math>
where <math> L^1_+(\mu)=\{g\in L^1(\mu): g\geq0\} </math>. </blockquote>
Since Hunt's definition is equivalent to Definition A when the underlying measure space is finite (see Theorem 2 below), Definition H is widely adopted in Mathematics.
The following result<ref>{{cite book|first=A.|last=Klenke|year=2008|title=Probability Theory: A Comprehensive Course|publisher=Springer Verlag|location=Berlin|isbn= 978-1-84800-047-6|pages=134–137}}</ref> provides another equivalent notion to Hunt's. This equivalency is sometimes given as definition for uniform integrability.
<blockquote> '''Theorem 1:''' If <math> (X,\mathfrak{M},\mu)</math> is a (positive) finite measure space, then a set <math> \Phi\subset L^1(\mu)</math> is uniformly integrable if and only if
:<math> \inf_{g\in L^1_+(\mu)}\sup_{f\in\Phi}\int (|f|- g)^+ \, d\mu=0 </math>
If in addition <math>\mu(X)<\infty</math>, then uniform integrability is equivalent to either of the following conditions
1. <math>\inf_{a>0}\sup_{f\in \Phi}\int(|f|-a)_+\,d\mu =0</math>.
2. <math>\inf_{a>0}\sup_{f\in \Phi}\int_{\{|f|>a\}}|f|\,d\mu=0</math> </blockquote>
When the underlying space <math> (X,\mathfrak{M},\mu) </math> is <math> \sigma </math>-finite, Hunt's definition is equivalent to the following:
<blockquote> '''Theorem 2:''' Let <math> (X,\mathfrak{M},\mu)</math> be a <math> \sigma </math>-finite measure space, and <math> h\in L^1(\mu) </math> be such that <math> h>0 </math> almost everywhere. A set <math> \Phi\subset L^1(\mu)</math> is uniformly integrable if and only if <math> \sup_{f\in\Phi}\|f\|_{L^1(\mu)}<\infty </math>, and for any <math> \varepsilon>0 </math>, there exists <math> \delta>0 </math> such that
:<math> \sup_{f\in\Phi}\int_A|f|\, d\mu <\varepsilon </math>
whenever <math> \int_A h\,d\mu <\delta </math>. </blockquote>
A consequence of Theorems 1 and 2 is that equivalence of Definitions A and H for finite measures follows. Indeed, the statement in Definition A is obtained by taking <math> h\equiv1</math> in Theorem 2.
==Tightness, boundedness, equi-integrability and uniform integrability== Another concept associated with uniform integrability is that of '''tightness'''. In this article tightness is taken in a more general setting. <blockquote> '''Definition:''' Suppose <math>(X,\mathfrak{M},\mu)</math> is a measure space. Let <math>\mathcal{K}\subset\mathfrak{M}</math> be a collection of sets of finite measure. A family <math>\Phi\subset L^1(\mu)</math> is said to be '''tight with respect to''' <math>\mathcal{K}</math> if :<math> \inf_{K\in\mathcal{K}}\sup_{f\in\Phi}\int_{X\setminus K}|f|\,d\mu=0 </math> When <math>\mathcal{K}=\mathfrak{M}\cap L^1(\mu)</math>, <math>\Phi</math> is simply said to be '''tight'''. </blockquote>
When the measure space <math>(X,\mathfrak{M},\mu)</math> is a metric space equipped with the Borel <math>\sigma</math> algebra, <math>\mu</math> is a regular measure, and <math>\mathcal{K}</math> is the collection of all compact subsets of <math>X</math>, the notion of <math>\mathcal{K}</math>-tightness discussed above coincides with the well known concept of tightness used in the analysis of regular measures in metric spaces
For <math>\sigma</math>-finite measure spaces, it can be shown that if a family <math>\Phi\subset L^1(\mu)</math> is uniformly integrable, then <math>\Phi</math> is tight. This is captured by the following result which is often used as definition of uniform integrability in the analysis literature:
<blockquote> '''Theorem 3:''' Suppose <math> (X,\mathfrak{M},\mu)</math> is a <math>\sigma</math>-finite measure space. A family <math>\Phi\subset L^1(\mu)</math> is uniformly integrable if and only if # <math>\sup_{f\in\Phi}\|f\|_1<\infty</math>. # <math>\inf_{a>0}\sup_{f\in \Phi}\int_{\{|f|>a\}}|f|\,d\mu=0</math> # <math>\Phi</math> is tight.
When <math>\mu(X)<\infty</math>, condition 3 is redundant (see Theorem 1 above). </blockquote>
In many books in analysis <ref>{{cite book |last1=Fonseca |first1=Irene |last2=Leoni |first2=Giovanni |title=Modern Methods in the Calculus of Variations: Lp Spaces |date=2007 |publisher=Springer New York Springer e-books |location=New York, NY |isbn=978-0387690063}}</ref><ref>{{cite book|first=J. J. |last=Benedetto|authorlink=J. J. Benedetto |year=1976|title=Real Variable and Integration|publisher=B. G. Teubner |location=Stuttgart|page=89| isbn=3-519-02209-5}}</ref><ref>{{cite book |first=C. W.|last=Burrill|authorlink=C. W. Burrill|year=1972|title=Measure, Integration, and Probability| publisher=McGraw-Hill|page=180| isbn=0-07-009223-0}}</ref><ref>{{cite book|last=Bass|first=Richard F.|title=Stochastic Processes|year=2011|publisher=Cambridge University Press|location=Cambridge|isbn=978-1-107-00800-7|pages=356–357}}</ref>, condition 2 in Theorem 3 is often replaced by another condition called '''equi-integrability''':
<blockquote> '''Definition:''' A family <math>\mathcal{C}</math> of complex or real valued measurable functions is '''equi-integrable''' (or '''uniformly absolutely continuous''' with respect to a measure <math>\mu</math>) if for any <math>\varepsilon>0</math> there is <math>\delta>0</math> such that <math display="block">\sup_{f\in\mathcal{C}}\int_A|f|\,d\mu<\varepsilon \qquad\text{whenever}\qquad \mu(A)<\delta</math> </blockquote> Theorem 3 then says that '''equi-integrability''' together with <math>L^1</math> '''boundedness''' and '''tightness''' (conditions (1) and (3) in Theorem 3) is equivalent to '''uniform integrability'''.
==Relevant theorems== The following theorems describe very useful criteria for uniform integrability which have many applications in Analysis and Probability. <blockquote>'''de la Vallée-Poussin theorem'''<ref>Meyer, P.A. (1966). ''Probability and Potentials'', Blaisdell Publishing Co, N. Y. (p.19, Theorem T22).</ref><ref>{{Cite journal|last=De La Vallée Poussin | first=C. |author-link1=Charles Jean de la Vallée-Poussin|date=1915|title=Sur L'Integrale de Lebesgue | journal=Transactions of the American Mathematical Society|volume=16 |issue=4 |pages=435–501 |doi=10.2307/1988879 |jstor=1988879 |hdl=10338.dmlcz/127627 |hdl-access=free}}</ref>{{pb}} Suppose <math>(X,\mathfrak{M},\mu)</math> is a finite measure space. The family <math>\mathcal{F} \subset L^1(\mu)</math> is uniformly integrable if and only if there exists a function <math>G:[0,\infty)\rightarrow[0,\infty)</math> such that <math>\lim_{t \to \infty} \frac{G(t)} t = \infty </math> and <math display="block"> \sup_{f\in\mathcal{F}} \int_X G(|f|)\,d\mu < \infty.</math> The function <math>G</math> can be chosen to be monotone increasing and convex.</blockquote>
Uniform integrability gives a characterization of weak compactness in <math>L^1</math>.
<blockquote>'''Dunford–Pettis theorem'''<ref>{{Cite journal|last=Dunford|first=Nelson|date=1938|title=Uniformity in linear spaces |url=https://www.ams.org/| journal=Transactions of the American Mathematical Society |language=en |volume=44 |issue=2|pages=305–356 |doi=10.1090/S0002-9947-1938-1501971-X |issn=0002-9947 | doi-access=free}}</ref><ref>{{Cite journal | last=Dunford|first=Nelson |date=1939 | title=A mean ergodic theorem| journal=Duke Mathematical Journal |language=en|volume=5 |issue=3|pages=635–646|doi=10.1215/S0012-7094-39-00552-1|issn=0012-7094}}</ref>{{pb}} Suppose <math>(X,\mathfrak{M},\mu)</math> is a <math>\sigma</math>-finite measure. A family <math>\mathcal{F}\subset L^1(\mu)</math> has compact closure in the weak topology <math>\sigma(L^1,L^\infty)</math> if and only if <math>\mathcal{F}</math> is uniformly integrable. </blockquote>
==Probability definition== In probability theory, Definition A or the statement of Theorem 1 are often presented as definitions of uniform integrability using the notation expectation of random variables.,<ref>{{cite book|last=Williams|first=David|title=Probability with Martingales|year=1997|publisher=Cambridge Univ. Press.|location=Cambridge|isbn=978-0-521-40605-5|pages=126–132|edition=Repr.}}</ref><ref>{{cite book|last=Gut|first=Allan|title=Probability: A Graduate Course|year=2005|publisher=Springer|isbn=0-387-22833-0|pages=214–218}}</ref><ref>{{cite book|last=Bass|first=Richard F.|title=Stochastic Processes|year=2011|publisher=Cambridge University Press|location=Cambridge|isbn=978-1-107-00800-7|pages=356–357}}</ref> that is,
1. A class <math>\mathcal{C}</math> of random variables is called '''uniformly integrable''' if:
* There exists a finite <math>M</math> such that, for every <math>X</math> in <math>\mathcal{C}</math>, <math>\operatorname E(|X|)\leq M</math> and * For every <math>\varepsilon > 0</math> there exists <math>\delta > 0</math> such that, for every measurable <math>A</math> such that <math>P(A)\leq \delta</math> and every <math>X</math> in <math>\mathcal{C}</math>, <math>\operatorname E(|X|I_A)\leq\varepsilon</math>.
or alternatively
2. A class <math>\mathcal{C}</math> of random variables is called '''uniformly integrable''' (UI) if for every <math>\varepsilon > 0</math> there exists <math>K\in[0,\infty)</math> such that <math>\operatorname E(|X|I_{|X|\geq K})\le\varepsilon\ \text{ for all } X \in \mathcal{C}</math>, where <math> I_{|X|\geq K} </math> is the indicator function <math> I_{|X|\geq K} = \begin{cases} 1 &\text{if } |X|\geq K, \\ 0 &\text{if } |X| < K. \end{cases}</math>.
==Related corollaries== The following results apply to the probabilistic definition.{{sfn|Gut|2005|pages=215–216}} * Definition 1 could be rewritten by taking the limits as <math display="block">\lim_{K \to \infty} \sup_{X \in \mathcal{C}} \operatorname E(|X|\,I_{|X|\geq K})=0.</math> * A non-UI sequence. Let <math>\Omega = [0,1] \subset \mathbb{R}</math>, and define <math display="block">X_n(\omega) = \begin{cases} n, & \omega\in (0,1/n), \\ 0 , & \text{otherwise.} \end{cases}</math> Clearly <math>X_n\in L^1</math>, and indeed <math>\operatorname E(|X_n|)=1\ ,</math> for all ''n''. However, <math display="block">\operatorname E(|X_n| I_{\{|X_n|\ge K \}})= 1\ \text{ for all } n \ge K,</math> and comparing with definition 1, it is seen that the sequence is not uniformly integrable. thumb|Non-UI sequence of RVs. The area under the strip is always equal to 1, but <math>X_n \to 0</math> pointwise. * By using Definition 2 in the above example, it can be seen that the first clause is satisfied as <math>L^1</math> norm of all <math>X_n</math>s are 1 i.e., bounded. But the second clause does not hold as given any <math>\delta </math> positive, there is an interval <math> (0, 1/n)</math> with measure less than <math>\delta</math> and <math>E[|X_m|: (0, 1/n)] =1 </math> for all <math>m \ge n </math>. * If <math>X</math> is a '''UI''' random variable, by splitting <math display="block">\operatorname E(|X|) = \operatorname E(|X| I_{\{|X| \geq K \}})+\operatorname E(|X| I_{\{|X| < K \}})</math> and bounding each of the two, it can be seen that a uniformly integrable random variable is always bounded in <math>L^1</math>. * If any sequence of random variables <math>X_n</math> is dominated by an integrable, non-negative <math>Y</math>: that is, for all ''ω'' and ''n'', <math display="block"> |X_n(\omega)| \le Y(\omega),\ Y(\omega)\ge 0,\ \operatorname E(Y) < \infty,</math> then the class <math>\mathcal{C}</math> of random variables <math>\{X_n\}</math> is uniformly integrable. * A class of random variables bounded in <math>L^p</math> (<math>p > 1</math>) is uniformly integrable.
==Uniform integrability and stochastic ordering== A family of random variables <math>\{X_i\}_{i \in I}</math> is uniformly integrable if and only if<ref>{{Cite journal|author1=Leskelä, L.|author2=Vihola, M. | date=2013 |title=Stochastic order characterization of uniform integrability and tightness |url=https://www.sciencedirect.com/science/article/abs/pii/S0167715212003690?via%3Dihub| journal=Statistics and Probability Letters | language=en | volume=83 | issue=1 | pages=382–389 | doi=10.1016/j.spl.2012.09.023| arxiv=1106.0607 }}</ref> there exists a random variable <math>X</math> such that <math>E X < \infty</math> and <math> |X_i| \le_\mathrm{icx} X</math> for all <math>i \in I</math>, where <math>\le_\mathrm{icx}</math> denotes the increasing convex stochastic order defined by <math>A \le_\mathrm{icx} B</math> if <math>E \phi(A) \le E \phi(B)</math> for all nondecreasing convex real functions <math>\phi</math>.
==Relation to convergence of random variables== {{main|Convergence of random variables}} A sequence <math>\{X_n\}</math> converges to <math>X</math> in the <math>L^1</math> norm if and only if it converges in measure to <math>X</math> and it is uniformly integrable. In probability terms, a sequence of random variables converging in probability also converge in the mean if and only if they are uniformly integrable.<ref>{{cite book|last=Bogachev|first=Vladimir I.|chapter=The spaces Lp and spaces of measures | title=Measure Theory Volume I|year=2007 |publisher=Springer-Verlag|location=Berlin Heidelberg|isbn=978-3-540-34513-8|pages=268|doi=10.1007/978-3-540-34514-5_4}}</ref> This is a generalization of Lebesgue's dominated convergence theorem, see Vitali convergence theorem.
==Citations== {{Reflist}}
==References== * {{cite book|authorlink=Albert Nikolayevich Shiryaev|first=A.N.|last=Shiryaev|year=1995|title=Probability|edition=2|publisher=Springer-Verlag|location=New York|pages=187–188|isbn=978-0-387-94549-1}} * Diestel, J. and Uhl, J. (1977). ''Vector measures'', Mathematical Surveys 15, American Mathematical Society, Providence, RI {{isbn|978-0-8218-1515-1}}
{{Stochastic processes}}
Category:Martingale theory