Uniform integrability

The concept of uniform integrability is an important concept in functional analysis and probability theory.

If $\mu$ is a probability mesure, a subset $K\subset L^{1}(\mu )$ is said to be uniformly integrable if $\lim _{c\to \infty }\sup _{X\in K}\int _{|X|\geq c}|X|d\mu =0$.

Rephrased with a probabilistic language, the definition becomes : a family $\{X_{\alpha }\}_{\alpha \in \mathrm {A} }$ of integrable random variables is uniformly integrable if

$\sup _{\alpha }\mathrm {E} \left[|X_{\alpha }|I_{\{|X_{\alpha }|>c\}}\right]\to 0,\;c\to \infty$.

This definition is useful in limit theorems, such as Lévy's convergence theorem.

Sufficient conditions

• If $X\in L^{1}(\mu )$, the singleton $\{X\}$ is uniformly integrable, as an easy consequence of Lebesgue's dominated convergence theorem. Similarly, finite subsets of $L^{1}(\mu )$ are uniformly integrable.

• If $X\in L^{1}(\mu )$, the set $\{Y\in L^{1}(\mu )s.t.|Y|\leq X\}$ is uniformly integrable.

Sufficient and necessary conditions

• A subset $K\subset L^{1}(\mu )$ is uniformly integrable iff it is uniformly bounded (i.e. $\sup _{X\in K}\|X\|_{L^{1}(\mu )}<\infty$) and absolutely continuous, i.e. for any $\epsilon >0$ there exists $\delta >0$ so that

$\mu (A)\leq \delta \Longrightarrow \sup _{X\in K}\int _{A}|X|d\mu \leq \epsilon$.

• The Dunford-Pettis theorem asserts that a subset $K\subset L^{1}(\mu )$ is uniformly integrable if and only if it is relatively compact for the weak topology.

• (Vallée-Poussin) The family $\{X_{\alpha }\}_{\alpha \in \mathrm {A} }$ is uniformly integrable iff there exists a nonnegative increasing function $G(t)$ such that $\lim _{t}{\frac {G(t)}{t}}=\infty$ and $\sup _{\alpha }E(G(|X_{\alpha }|))<\infty$

Relations to convergence of random variables

• Let $(X_{n})$ be a sequence of integrable random variables. If $(X_{n})$ converges in probability to $X$ and is uniformly integrable, then $(X_{n})$ converges to $X$ in the $L_{1}$ norm.

References

• A.N.Shiryaev (1995). Probability, 2nd Edition, Springer-Verlag, New York, pp.187-188, ISBN 978-0387945491

• Walter Rudin (1987). Real and Complex Analysis, 3rd Edition, McGraw-Hill Book Co., Singapore, pp.133, ISBN 0-07-054234-1

• J. Diestel and J. Uhl, Vector measres, Mathematical Surveys 15 (1977).