Uniform integrability

Uniform integrability is an important concept in real analysis, functional analysis and measure theory, and plays a vital role in the theory of martingales.

Definition

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

$|\int _{E}fd\mu |<\epsilon$

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

Formal definition

The following definition applies.

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

• An alternative definition involving two clauses may be presented as follows: A class ${\mathcal {C}}$ of random variables is called uniformly integrable if:
  • There exists a finite $K$ such that, for every $X$ in ${\mathcal {C}}$, $\mathrm {E} (|X|)\leqslant K$.
  • For every $\epsilon >0$ there exists $\delta >0$ such that, for every measurable $A$ such that $\mathrm {P} (A)\leqslant \delta$ and every $X$ in ${\mathcal {C}}$, $\mathrm {E} (|X|:A)\leqslant \epsilon$.

Related corollaries

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• Definition 1 could be rewritten by taking the limits as

$\lim _{K\to \infty }\sup _{X\in {\mathcal {C}}}E(|X|I_{|X|\geq K})=0.$

• 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 $E(|X_{n}|)=1\ ,$ for all n. However,

$E(|X_{n}|,|X_{n}|\geq K)=1\ {\text{ for all }}n\geq K,$

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

$E(|X|)=E(|X|,|X|>K)+E(|X|,|X|<K)$

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,

$\ |X_{n}(\omega )|\leq |Y(\omega )|,\ Y(\omega )\geq 0,\ E(Y)<\infty ,$

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.

Relevant theorems

• Dunford–Pettis theorem

A class of random variables $X_{n}\subset L^{1}(\mu )$ is uniformly integrable if and only if it is relatively compact for the weak topology $\sigma (L^{1},L^{\infty })$.

• de la Vallée-Poussin theorem

The family $\{X_{\alpha }\}_{\alpha \in \mathrm {A} }\subset L^{1}(\mu )$ is uniformly integrable if and only if there exists a non-negative increasing convex function $G(t)$ such that

$\lim _{t\to \infty }{\frac {G(t)}{t}}=\infty$ and $\sup _{\alpha }E(G(|X_{\alpha }|))<\infty .$

Relation to convergence of random variables

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• 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 the dominated convergence theorem.

References

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• J. Diestel and J. Uhl (1977). Vector measures, Mathematical Surveys 15, American Mathematical Society, Providence, RI ISBN 978-0-8218-1515-1