# 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

$\left| \int_E f d\mu \right| < \epsilon$

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

## Formal definition

The following definition applies.[1]

• A class $\mathcal{C}$ of random variables is called uniformly integrable (UI) if given $\epsilon>0$, there exists $K\in[0,\infty)$ such that $E(|X|I_{|X|\geq K})\le\epsilon\ \text{ for all X} \in \mathcal{C}$, 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

The following results apply.[citation needed]

• 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,
$E(|X_n|,|X_n|\ge K)= 1\ \text{ for all } n\ge K,$
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 $m \ge n$.
• If $X$ is a UI random variable, by splitting
$E(|X|)=E(|X|,|X|>K)+E(|X|,|X|
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,
$\ |X_n(\omega)| \le |Y(\omega)|,\ Y(\omega)\ge 0,\ E(Y)< \infty,$
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

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 $\sigma(L^1,L^\infty)$.
The family $\{X_{\alpha}\}_{\alpha\in\Alpha} \subset L^1(\mu)$ 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

• 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.[4] This is a generalization of the dominated convergence theorem.

## Citations

1. ^ Williams, David (1997). Probability with Martingales (Repr. ed.). Cambridge: Cambridge Univ. Press. pp. 126–132. ISBN 978-0-521-40605-5.
2. ^ Dellacherie, C. and Meyer, P.A. (1978). Probabilities and Potential, North-Holland Pub. Co, N. Y. (Chapter II, Theorem T25).
3. ^ Meyer, P.A. (1966). Probability and Potentials, Blaisdell Publishing Co, N. Y. (p.19, Theorem T22).
4. ^ Bogachev, Vladimir I. (2007). Measure Theory Volume I. Berlin Heidelberg: Springer-Verlag. p. 268. doi:10.1007/978-3-540-34514-5_4. ISBN 3-540-34513-2.