Uniform integrability

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 $L_{1}$ which is central in dominated convergence. Several textbooks on real analysis and measure theory use the following definition:

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

$\int _{E}|f|\,d\mu <\varepsilon$

whenever $f\in \Phi$ and $\mu (E)<\delta .$

Definition A is rather restrictive for infinite measure spaces. A more general definition of uniform integrability that works well in general measure spaces was introduced by G. A. Hunt.

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

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

where $L_{+}^{1}(\mu )=\{g\in L^{1}(\mu ):g\geq 0\}$.

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

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

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

If in addition $\mu (X)<\infty$, then uniform integrability is equivalent to either of the following conditions

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

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

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

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

$\sup _{f\in \Phi }\int _{A}|f|\,d\mu <\varepsilon$

whenever $\int _{A}h\,d\mu <\delta$.

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 $h\equiv 1$ 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.

Definition: Suppose measurable space $(X,{\mathfrak {M}},\mu )$ is a measure space. Let ${\mathcal {K}}\subset {\mathfrak {M}}$ be a collection of sets of finite measure. A family $\Phi \subset L_{1}(\mu )$ is said to be tight with respect to ${\mathcal {K}}$ if

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

When ${\mathcal {K}}={\mathfrak {M}}\cap L_{1}(\,u)$, $\Phi$ is simply said to be tight.

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

For $\sigma$-finite measure spaces, it can be shown that if a family $\Phi \subset L_{1}(\mu )$ is uniformly integrable, then $\Phi$ 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 $(X,{\mathfrak {M}},\mu )$ is a $\sigma$ finite measure space. A family $\Phi \subset L_{1}(\mu )$ is uniformly integrable if and only if

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

When $\mu (X)<\infty$, condition 3 is redundant (see Theorem 1 above).

In many books in Analysis , condition 2 in Theorem 3 is often replaced by another condition called equi-integrability:

Definition: A family ${\mathcal {C}}$ of complex or real valued measurable functions is equi-integrable (or uniformly absolutely continuous with respect to a measure $\mu$) if for any $\varepsilon >0$ there is $\delta >0$ such that $$\sup _{f\in {\mathcal {C}}}\int _{A}|f|\,d\mu \qquad {\text{whenever}}\qquad \mu (A)<\delta$$

Theorem 3 then says that equi-integrability together with $L_{1}$ 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.

de la Vallée-Poussin theorem

Suppose $(X,{\mathfrak {M}},\mu )$ is a finite measure space. The family ${\mathcal {F}}\subset L^{1}(\mu )$ is uniformly integrable if and only if there exists a function $G:[0,\infty )\rightarrow [0,\infty )$ such that $\lim _{t\to \infty }{\frac {G(t)}{t}}=\infty$ and $$\sup _{f\in {\mathcal {F}}}\int _{X}G(|f|)\,d\mu <\infty .$$ The function $G$ can be chosen to be monotone increasing and convex.

Uniform integrability gives a characterization of weak compactness in $L_{1}$.

Dunford–Pettis theorem

Suppose $(X,{\mathfrak {M}},\mu )$ is a $\sigma$-finite measure. A family ${\mathcal {F}}\subset L_{1}(\mu )$ has compact closure in the weak topology $\sigma (L_{1},L_{\infty })$ if and only if ${\mathcal {F}}$ is uniformly integrable.

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., that is,

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

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

or alternatively

2. A class ${\mathcal {C}}$ of random variables is called uniformly integrable (UI) if for every $\varepsilon >0$ there exists $K\in [0,\infty )$ such that $\operatorname {E} (|X|I_{|X|\geq K})\leq \varepsilon \ {\text{ for all }}X\in {\mathcal {C}}$, where $I_{|X|\geq K}$ is the indicator function $I_{|X|\geq K}={\begin{cases}1&{\text{if }}|X|\geq K,\\0&{\text{if }}|X|<K.\end{cases}}$.

Related corollaries

The following results apply to the probabilistic definition.

• Definition 1 could be rewritten by taking the limits as $$\lim _{K\to \infty }\sup _{X\in {\mathcal {C}}}\operatorname {E} (|X|\,I_{|X|\geq K})=0.$$
• A non-UI sequence. Let $\Omega =[0,1]\subset \mathbb {R}$, and define $$X_{n}(\omega )={\begin{cases}n,&\omega \in (0,1/n),\\0,&{\text{otherwise.}}\end{cases}}$$ Clearly $X_{n}\in L^{1}$, and indeed $\operatorname {E} (|X_{n}|)=1\ ,$ for all n. However, $$\operatorname {E} (|X_{n}|I_{\{|X_{n}|\geq K\}})=1\ {\text{ for all }}n\geq K,$$ and comparing with definition 1, it is seen that the sequence is not uniformly integrable.

Non-UI sequence of RVs. The area under the strip is always equal to 1, but $X_{n}\to 0$ pointwise.

• 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 $\delta$ positive, there is an interval $(0,1/n)$ with measure less than $\delta$ and $E[|X_{m}|:(0,1/n)]=1$ for all $m\geq n$.
• If $X$ is a UI random variable, by splitting $$\operatorname {E} (|X|)=\operatorname {E} (|X|I_{\{|X|\geq K\}})+\operatorname {E} (|X|I_{\{|X|<K\}})$$ 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, $$|X_{n}(\omega )|\leq Y(\omega ),\ Y(\omega )\geq 0,\ \operatorname {E} (Y)<\infty ,$$ then the class ${\mathcal {C}}$ of random variables $\{X_{n}\}$ is uniformly integrable.
• A class of random variables bounded in $L^{p}$ ($p>1$) is uniformly integrable.

Uniform integrability and stochastic ordering

A family of random variables $\{X_{i}\}_{i\in I}$ is uniformly integrable if and only if there exists a random variable $X$ such that $EX<\infty$ and $|X_{i}|\leq _{\mathrm {icx} }X$ for all $i\in I$, where $\leq _{\mathrm {icx} }$ denotes the increasing convex stochastic order defined by $A\leq _{\mathrm {icx} }B$ if $E\phi (A)\leq E\phi (B)$ for all nondecreasing convex real functions $\phi$.

Relation to convergence of random variables

Main article: Convergence of random variables

A sequence $\{X_{n}\}$ 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. This is a generalization of Lebesgue's dominated convergence theorem, see Vitali convergence theorem.

References

• Shiryaev, A.N. (1995). Probability (2 ed.). New York: Springer-Verlag. pp. 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