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

## Formal definition

The following definition applies.[1]

## Related corollaries

The following results apply.{{ safesubst:#invoke:Unsubst||date=__DATE__ |\$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }}

• Definition 1 could be rewritten by taking the limits as
${\displaystyle \lim _{K\to \infty }\sup _{X\in {\mathcal {C}}}E(|X|I_{|X|\geq K})=0.}$
${\displaystyle X_{n}(\omega )={\begin{cases}n,&\omega \in (0,1/n),\\0,&{\text{otherwise.}}\end{cases}}}$
Clearly ${\displaystyle X_{n}\in L^{1}}$, and indeed ${\displaystyle E(|X_{n}|)=1\ ,}$ for all n. However,
${\displaystyle E(|X_{n}|,|X_{n}|\geq K)=1\ {\text{ for all }}n\geq K,}$
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 ${\displaystyle X_{n}\to 0}$ pointwise.
${\displaystyle 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 ${\displaystyle L^{1}}$.
${\displaystyle \ |X_{n}(\omega )|\leq |Y(\omega )|,\ Y(\omega )\geq 0,\ E(Y)<\infty ,}$
then the class ${\displaystyle {\mathcal {C}}}$ of random variables ${\displaystyle \{X_{n}\}}$ is uniformly integrable.

## Relevant theorems

A class of random variables ${\displaystyle X_{n}\subset L^{1}(\mu )}$ is uniformly integrable if and only if it is relatively compact for the weak topology ${\displaystyle \sigma (L^{1},L^{\infty })}$.
The family ${\displaystyle \{X_{\alpha }\}_{\alpha \in \mathrm {A} }\subset L^{1}(\mu )}$ is uniformly integrable if and only if there exists a non-negative increasing convex function ${\displaystyle G(t)}$ such that
${\displaystyle \lim _{t\to \infty }{\frac {G(t)}{t}}=\infty }$ and ${\displaystyle \sup _{\alpha }E(G(|X_{\alpha }|))<\infty .}$

## Relation to convergence of random variables

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## Citations

1. {{#invoke:citation/CS1|citation |CitationClass=book }}
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. {{#invoke:citation/CS1|citation |CitationClass=book }}

## 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