Uniform integrability

Uniform integrability is an extension to the notion of a family of functions being dominated in $L\_1$ which is central in dominated convergence.

Several textbooks on real analysis and measure theory use the following definition:{{cite book|first=Walter|last=Rudin|authorlink=Walter Rudin|year=1987|title=Real and Complex Analysis|edition=3|publisher=McGraw–Hill Book Co.|location=Singapore|page=133|isbn=0-07-054234-1}}{{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}}

Definition A: Let $(X,\backslash mathfrak\{M\},\; \backslash mu)$ be a positive measure space. A set $\backslash Phi\backslash subset\; L^1(\backslash mu)$ is called uniformly integrable if , and to each $\backslash varepsilon>0$ there corresponds a $\backslash delta>0$ such that

:

whenever $f\; \backslash in\; \backslash Phi$ and

Definition A is rather restrictive for infinite measure spaces. A more general definition{{cite book|first=G. A.|last=Hunt|year=1966|title=Martingales et Processus de Markov|publisher=Dunod|location=Paris|page=254}} of uniform integrability that works well in general measures spaces was introduced by G. A. Hunt.

Definition H: Let $(X,\backslash mathfrak\{M\},\backslash mu)$ be a positive measure space. A set $\backslash Phi\backslash subset\; L^1(\backslash mu)$ is called uniformly integrable if and only if

:$\backslash inf\_\{g\backslash in\; L^1\_+(\backslash mu)\}\backslash sup\_\{f\backslash in\backslash Phi\}\backslash int\_\{\backslash \{|f|>g\backslash \}\}|f|\backslash ,\; d\backslash mu=0$

where $L^1\_+(\backslash mu)=\backslash \{g\backslash in\; L^1(\backslash mu):\; g\backslash geq0\backslash \}$.

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{{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}} provides another equivalent notion to Hunt's. This equivalency is sometimes given as definition for uniform integrability.

Theorem 1: If $(X,\backslash mathfrak\{M\},\backslash mu)$ is a (positive) finite measure space, then a set $\backslash Phi\backslash subset\; L^1(\backslash mu)$ is uniformly integrable if and only if

:$\backslash inf\_\{g\backslash in\; L^1\_+(\backslash mu)\}\backslash sup\_\{f\backslash in\backslash Phi\}\backslash int\; (|f|-\; g)^+\; \backslash ,\; d\backslash mu=0$

If in addition , then uniform integrability is equivalent to either of the following conditions

1. $\backslash inf\_\{a>0\}\backslash sup\_\{f\backslash in\; \backslash Phi\}\backslash int(|f|-a)\_+\backslash ,d\backslash mu\; =0$.

2. $\backslash inf\_\{a>0\}\backslash sup\_\{f\backslash in\; \backslash Phi\}\backslash int\_\{\backslash \{|f|>a\backslash \}\}|f|\backslash ,d\backslash mu=0$

When the underlying space $(X,\backslash mathfrak\{M\},\backslash mu)$ is $\backslash sigma$-finite, Hunt's definition is equivalent to the following:

Theorem 2: Let $(X,\backslash mathfrak\{M\},\backslash mu)$ be a $\backslash sigma$-finite measure space, and $h\backslash in\; L^1(\backslash mu)$ be such that $h>0$ almost everywhere. A set $\backslash Phi\backslash subset\; L^1(\backslash mu)$ is uniformly integrable if and only if , and for any $\backslash varepsilon>0$, there exits $\backslash delta>0$ such that

:

whenever .

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 $h\backslash equiv1$ in Theorem 2.

In the theory of probability, Definition A or the statement of Theorem 1 are often presented as definitions of uniform integrability using the notation expectation of random variables.,{{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.}}{{cite book|last=Gut|first=Allan|title=Probability: A Graduate Course|year=2005|publisher=Springer|isbn=0-387-22833-0|pages=214–218}}{{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}} that is,

1. A class $\backslash mathcal\{C\}$ of random variables is called uniformly integrable if:

• There exists a finite $M$ such that, for every $X$ in $\backslash mathcal\{C\}$, $\backslash operatorname\; E(|X|)\backslash leq\; M$ and
• For every $\backslash varepsilon\; >\; 0$ there exists $\backslash delta\; >\; 0$ such that, for every measurable $A$ such that $P(A)\backslash leq\; \backslash delta$ and every $X$ in $\backslash mathcal\{C\}$, $\backslash operatorname\; E(|X|I\_A)\backslash leq\backslash varepsilon$.

or alternatively

2. A class $\backslash mathcal\{C\}$ of random variables is called uniformly integrable (UI) if for every $\backslash varepsilon\; >\; 0$ there exists $K\backslash in[0,\backslash infty)$ such that $\backslash operatorname\; E(|X|I\_\{|X|\backslash geq\; K\})\backslash le\backslash varepsilon\backslash \; \backslash text\{\; for\; all\; \}\; X\; \backslash in\; \backslash mathcal\{C\}$, where $I\_\{|X|\backslash geq\; K\}$ is the indicator function .

Another concept associated with uniform integrability is that of tightness. In this article tightness is taken in a more general setting.

Definition: Suppose measurable space $(X,\backslash mathfrak\{M\},\backslash mu)$ is a measure space. Let $\backslash mathcal\{K\}\backslash subset\backslash mathfrak\{M\}$ be a collection of sets of finite measure. A family $\backslash Phi\backslash subset\; L\_1(\backslash mu)$ is tight with respect to $\backslash mathcal\{K\}$ if

:$\backslash inf\_\{K\backslash in\backslash mathcal\{K\}\}\backslash sup\_\{f\backslash in\backslash Phi\}\backslash int\_\{X\backslash setminus\; K\}|f|\backslash ,\backslash mu=0$

A tight family with respect to $\backslash Phi=\backslash mathfrak\{M\}\backslash cap\; L\_1(\backslash ,u)$ is just said to be tight.

When the measure space $(X,\backslash mathfrak\{M\},\backslash mu)$ is a metric space equipped with the Borel $\backslash sigma$ algebra, $\backslash mu$ is a regular measure, and $\backslash mathcal\{K\}$ is the collection of all compact subsets of $X$, the notion of $\backslash mathcal\{K\}$-tightness discussed above coincides with the well known concept of tightness used in the analysis of regular measures in metric spaces

For $\backslash sigma$-finite measure spaces, it can be shown that if a family $\backslash Phi\backslash subset\; L\_1(\backslash mu)$ is uniformly integrable, then $\backslash Phi$ is tight. This is capture by the following result which is often used as definition of uniform integrabiliy in the Analysis literature:

Theorem 3: Suppose $(X,\backslash mathfrak\{M\},\backslash mu)$ is a $\backslash sigma$ finite measure space. A family $\backslash Phi\backslash subset\; L\_1(\backslash mu)$ is uniformly integrable if and only if

1. .
2. $\backslash inf\_\{a>0\}\backslash sup\_\{f\backslash in\; \backslash Phi\}\backslash int\_\{\backslash \{|f|>a\backslash \}\}|f|\backslash ,d\backslash mu=0$
3. $\backslash Phi$ is tight.

When , condition 3 is redundant (see Theorem 1 above).

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{{sfn|Bass|2011|page=356}}

Definition: Suppose $(\backslash Omega,\backslash mathcal\{F\},P)$ is a probability space. A class $\backslash mathcal\{C\}$ of random variables is uniformly absolutely continuous with respect to $P$ if for any $\backslash varepsilon>0$, there is $\backslash delta>0$ such that

whenever .

It is equivalent to uniform integrability if the measure is finite and has no atoms.

The term "uniform absolute continuity" is not standard,{{citation needed|date=August 2023}} but is used by some authors.{{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}}{{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}}

The following results apply to the probabilistic definition.{{sfn|Gut|2005|pages=215–216}}

• Definition 1 could be rewritten by taking the limits as $$\backslash lim\_\{K\; \backslash to\; \backslash infty\}\; \backslash sup\_\{X\; \backslash in\; \backslash mathcal\{C\}\}\; \backslash operatorname\; E(|X|\backslash ,I\_\{|X|\backslash geq\; K\})=0.$$
• A non-UI sequence. Let $\backslash Omega\; =\; [0,1]\; \backslash subset\; \backslash mathbb\{R\}$, and define $$X\_n(\backslash omega)\; =\; \backslash begin\{cases\}$$

n, & \omega\in (0,1/n), \\

0 , & \text{otherwise.} \end{cases} Clearly $X\_n\backslash in\; L^1$, and indeed $\backslash operatorname\; E(|X\_n|)=1\backslash \; ,$ for all n. However, $$\backslash operatorname\; E(|X\_n|\; I\_\{\backslash \{|X\_n|\backslash ge\; K\; \backslash \}\})=\; 1\backslash \; \backslash text\{\; for\; all\; \}\; n\; \backslash ge\; K,$$ and comparing with definition 1, it is seen that the sequence is not uniformly integrable.

File:Uniform integrability.png

• By using Definition 2 in the above example, it can be seen that the first clause is satisfied as $L^1$ norm of all $X\_n$s are 1 i.e., bounded. But the second clause does not hold as given any $\backslash delta$ positive, there is an interval $(0,\; 1/n)$ with measure less than $\backslash delta$ and $E[|X\_m|:\; (0,\; 1/n)]\; =1$ for all $m\; \backslash ge\; n$.
• If $X$ is a UI random variable, by splitting and bounding each of the two, it can be seen that a uniformly integrable random variable is always bounded in $L^1$.
• If any sequence of random variables $X\_n$ is dominated by an integrable, non-negative $Y$: that is, for all ω and n, then the class $\backslash mathcal\{C\}$ of random variables $\backslash \{X\_n\backslash \}$ is uniformly integrable.
• A class of random variables bounded in $L^p$ ($p\; >\; 1$) is uniformly integrable.

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 $L^1(\backslash mu)$.

• Dunford–Pettis theorem{{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}}{{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}}{{pb}}A class{{what|reason=What is X_n? A set, a sequence, a single random variable?|date=August 2023}} of random variables $X\_n\; \backslash subset\; L^1(\backslash mu)$ is uniformly integrable if and only if it is relatively compact for the weak topology $\backslash sigma(L^1,L^\backslash infty)$.{{what|reason=This notation is not explained in the given link.|date=August 2023}}{{citation needed|date=August 2023}}
• de la Vallée-Poussin theoremMeyer, P.A. (1966). Probability and Potentials, Blaisdell Publishing Co, N. Y. (p.19, Theorem T22).{{Cite journal|last=Poussin | first=C. De La Vallee|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}}{{pb}}The family $\backslash \{X\_\{\backslash alpha\}\backslash \}\_\{\backslash alpha\backslash in\backslash Alpha\}\; \backslash subset\; L^1(\backslash mu)$ is uniformly integrable if and only if there exists a non-negative increasing convex function $G(t)$ such that

A family of random variables $\backslash \{X\_i\backslash \}\_\{i\; \backslash in\; I\}$ is uniformly integrable if and only if{{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 }}

there exists a random variable

$X$ such that and

$|X\_i|\; \backslash le\_\backslash mathrm\{icx\}\; X$ for all $i\; \backslash in\; I$, where

$\backslash le\_\backslash mathrm\{icx\}$ denotes the increasing convex stochastic order defined by

$A\; \backslash le\_\backslash mathrm\{icx\}\; B$ if $E\; \backslash phi(A)\; \backslash le\; E\; \backslash phi(B)$ for all nondecreasing convex real functions $\backslash phi$.

{{main|Convergence of random variables}}

A sequence $\backslash \{X\_n\backslash \}$ converges to $X$ in the $L\_1$ norm if and only if it converges in measure to $X$ 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.{{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}} This is a generalization of Lebesgue's dominated convergence theorem, see Vitali convergence theorem.

{{Reflist}}

• {{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}}

Category:Martingale theory