# Uniform integrability
**Uniform integrability** is a property of a family of integrable random variables (or measurable functions) requiring that their integrals over the tails be small uniformly across the family: the family \( (X_i)_{i \in I} \) is uniformly integrable when
\[ \sup_{i \in I} \mathbb{E}\bigl(|X_i|\,\mathbf{1}_{\{|X_i| > c\}}\bigr) \to 0 \quad \text{as } c \to \infty. \]
It is the condition that upgrades convergence in probability to convergence in \( L^1 \), and it is the hinge of the convergence theory of martingales.[1](https://www.math.uwaterloo.ca/~dlmcleis/s901/chapt7.pdf) • [2](https://zackienzle.com/blog/uniform-integrability)
| Key fact | Detail |
|---|---|
| Definition | \( \sup_i \mathbb{E}(|X_i|\,\mathbf{1}_{\{|X_i|>c\}}) \to 0 \) as \( c \to \infty \)[1](https://www.math.uwaterloo.ca/~dlmcleis/s901/chapt7.pdf) |
| Consequence | Uniform integrability implies boundedness in \( L^1 \): \( \sup_i \mathbb{E}|X_i| < \infty \)[3](https://wt.iam.uni-bonn.de/fileadmin/WT/Inhalt/people/Andreas_Eberle/IntroStoAn1819/StochasticAnalysisComplete.pdf) • [4](https://adembo.su.domains/stat-310b/lnotes.pdf) |
| Sufficient conditions | Boundedness in \( L^p \) for some \( p > 1 \), or the entropy condition \( \sup_i \mathbb{E}[|X_i|(\log |X_i|)^+] < \infty \)[3](https://wt.iam.uni-bonn.de/fileadmin/WT/Inhalt/people/Andreas_Eberle/IntroStoAn1819/StochasticAnalysisComplete.pdf) |
| Vitali theorem | If \( X_n \to X \) in probability, then uniform integrability is equivalent to \( L^1 \) convergence, which is equivalent to \( \mathbb{E}|X_n| \to \mathbb{E}|X| \)[1](https://www.math.uwaterloo.ca/~dlmcleis/s901/chapt7.pdf) |
| Conditional expectations | For integrable \( X \), the family \( \{\mathbb{E}[X \mid \mathcal{F}] : \mathcal{F} \subseteq \mathcal{A} \} \) is uniformly integrable[3](https://wt.iam.uni-bonn.de/fileadmin/WT/Inhalt/people/Andreas_Eberle/IntroStoAn1819/StochasticAnalysisComplete.pdf) |
| Martingales | A martingale \( (X_n, \mathcal{F}_n) \) is uniformly integrable if and only if \( X_n = \mathbb{E}[X_\infty \mid \mathcal{F}_n] \), a Doob martingale[4](https://adembo.su.domains/stat-310b/lnotes.pdf) |
| Weak compactness | On a finite measure space, a subset of \( L^1 \) is relatively weakly compact if and only if it is uniformly integrable (Dunford–Pettis theorem)[5](https://www.math.toronto.edu/almut/Brezis.pdf) |
## Definition and basic properties
The definition says that the mass of each \( |X_i| \) above a high threshold \( c \) has expectation that can be made small simultaneously for all members of the family by taking \( c \) large enough. A single integrable random variable satisfies \( \mathbb{E}(|X|\,\mathbf{1}_{\{|X|>c\}}) \to 0 \) as \( c \to \infty \); uniform integrability asks that this tail decay be uniform over the index set, which is what a single dominating integrable variable would give but what mere individual integrability does not.[1](https://www.math.uwaterloo.ca/~dlmcleis/s901/chapt7.pdf) • [3](https://wt.iam.uni-bonn.de/fileadmin/WT/Inhalt/people/Andreas_Eberle/IntroStoAn1819/StochasticAnalysisComplete.pdf)
Several elementary facts follow directly.[1](https://www.math.uwaterloo.ca/~dlmcleis/s901/chapt7.pdf)
1. Any finite set of integrable random variables is uniformly integrable.
2. Any sequence that converges in \( L^1 \) is uniformly integrable.
3. Conversely, if a sequence converges almost surely and is uniformly integrable, then it converges in \( L^1 \).
4. Uniform integrability implies \( \sup_n \mathbb{E}|X_n| < \infty \).
The converse of the last point fails: boundedness in \( L^1 \) does not imply uniform integrability, and the standard counterexamples are sequences that put a fixed total amount of mass on events of vanishing probability.[3](https://wt.iam.uni-bonn.de/fileadmin/WT/Inhalt/people/Andreas_Eberle/IntroStoAn1819/StochasticAnalysisComplete.pdf) On a finite measure space, uniform integrability can equivalently be characterized as \( L^1 \) boundedness together with uniform absolute continuity of the integrals, the "epsilon-delta" condition that \( \sup_i \int_A |X_i| \, d\mu \) be small whenever the measure of \( A \) is small.[2](https://zackienzle.com/blog/uniform-integrability) • [6](https://djalil.chafai.net/blog/2014/03/09/de-la-vallee-poussin-on-uniform-integrability/)
## Sufficient criteria
The most used sufficient condition is a moment slightly stronger than first order. If \( \sup_{i} \mathbb{E}[|X_i|^p] < \infty \) for some \( p > 1 \), the family is uniformly integrable; so is a family satisfying the entropy condition \( \sup_i \mathbb{E}[|X_i|(\log |X_i|)^+] < \infty \).[3](https://wt.iam.uni-bonn.de/fileadmin/WT/Inhalt/people/Andreas_Eberle/IntroStoAn1819/StochasticAnalysisComplete.pdf) Both are instances of the criterion attributed to Charles-Jean Étienne Gustave Nicolas de la Vallée Poussin (1866–1962), the Belgian mathematician also known for his proof of the prime number theorem: if the family is bounded in \( L^\varphi \) for a non-decreasing function \( \varphi : \mathbb{R}_+ \to \mathbb{R}_+ \) with \( \varphi(x)/x \to \infty \) as \( x \to \infty \), such as \( x \mapsto x^p \) with \( p > 1 \) or \( x \mapsto x \log x \), then it is uniformly integrable.[6](https://djalil.chafai.net/blog/2014/03/09/de-la-vallee-poussin-on-uniform-integrability/) The same criterion shows that for any integrable \( X \) and any family of sub-\(\sigma\)-algebras \( (\mathcal{F}_i) \), the conditional expectations \( (\mathbb{E}[X \mid \mathcal{F}_i]) \) form a uniformly integrable family, via [Jensen's inequality](https://www.edgepedia.org/jensens-inequality) applied to \( \varphi \).[7](https://www.samuel-drapeau.info/SP_Lecture/lecture/03-Integration/036-uniform-integrability/)
## Vitali convergence theorem
Uniform integrability is the exact supplement that convergence in probability needs to become \( L^1 \) convergence. The Vitali convergence theorem states that if \( X_n \to X \) in probability, then the following are equivalent: \( \{X_n\} \) is uniformly integrable; \( X_n \to X \) in \( L^1 \); and \( \mathbb{E}|X_n| \to \mathbb{E}|X| \).[1](https://www.math.uwaterloo.ca/~dlmcleis/s901/chapt7.pdf) • [4](https://adembo.su.domains/stat-310b/lnotes.pdf) In this sense it is the sharp form of the dominated convergence theorem: domination by an integrable variable implies uniform integrability, but uniform integrability is the weaker, intrinsic condition.[8](https://www.ceremade.dauphine.fr/~forien/Nicolas_Forien_files/poly-prerentree-m2.pdf) The proof of the forward direction splits each \( |X_n - X| \) into a bounded part handled by the dominated convergence theorem and a tail part controlled by the uniform integrability, giving \( \limsup_n \mathbb{E}|X_n - X| \le \delta \) for every \( \delta > 0 \).[9](https://finmath.rutgers.edu/images/documents/finance/Martingales.pdf)
## Martingales
Uniform integrability is the standard hypothesis in martingale convergence theory. For a submartingale \( (X_n, \mathcal{F}_n) \), uniform integrability is equivalent to \( L^1 \) convergence, and either implies almost sure convergence to an integrable limit \( X_\infty \); for a martingale it further gives the closure relation \( X_n = \mathbb{E}(X_\infty \mid \mathcal{F}_n) \).[1](https://www.math.uwaterloo.ca/~dlmcleis/s901/chapt7.pdf) • [9](https://finmath.rutgers.edu/images/documents/finance/Martingales.pdf) Conversely, a uniformly integrable martingale is exactly a Doob martingale, that is, a process of the form \( X_n = \mathbb{E}[X_\infty \mid \mathcal{F}_n] \) for an integrable \( X_\infty \) and a filtration \( (\mathcal{F}_n) \); the uniform integrability of such a family follows from the conditional-expectation property above.[4](https://adembo.su.domains/stat-310b/lnotes.pdf) This identification underlies the martingale convergence theorem and applications such as optional stopping.[10](https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/UniformIntegrable.html)
The property also matters in more specialized settings. Criteria for uniform integrability of exponential martingales in the context of Markov processes play a central role in Girsanov transforms and in the study of spectral bounds of Feynman–Kac semigroups; one such criterion is, in general, weaker than the commonly used Novikov condition.[11](https://arxiv.org/abs/1105.3020)
## Weak compactness: the Dunford–Pettis theorem
Because \( L^1 \) is not reflexive, bounded sets in \( L^1 \) need not be relatively weakly compact, and uniform integrability is precisely what restores compactness. The Dunford–Pettis theorem states that on a finite measure space, a bounded subset \( \mathcal{F} \subset L^1 \) has compact closure in the [weak topology](https://www.edgepedia.org/weak-topology) \( \sigma(L^1, L^\infty) \) if and only if \( \mathcal{F} \) is uniformly integrable (equi-integrable).[5](https://www.math.toronto.edu/almut/Brezis.pdf) • [12](https://jordanbell.info/LaTeX/mathematics/dunford-pettis/) A third equivalent condition in the same circle of ideas is the de la Vallée Poussin condition: existence of a positive, convex, super-linearly increasing function \( G \) with \( \lim_{s \to \infty} G(s)/s = +\infty \) and \( \sup_{v} \int G(|v|) \, dt < \infty \).[13](https://www.numdam.org/article/ASNSP_2003_5_2_2_395_0.pdf) The theorem was obtained by [Nelson Dunford](https://www.edgepedia.org/nelson-dunford) and Billy James Pettis in 1940.[14](https://www.researchgate.net/publication/371943814_Weak_Compactness_Criterion_in_Wk_1_with_an_Existence_Theorem_of_Minimizers)
## Historical note
The criterion bearing de la Vallée Poussin's name goes back to his memoir *Sur l'intégrale de Lebesgue*, published in 1915 in the *Transactions of the American Mathematical Society* (volume 16, part 4, pages 435–501).[15](https://cir.nii.ac.jp/crid/1360292620414059776) In modern probability texts the topic typically appears as a dedicated chapter before the martingale convergence theorems, as in Williams's *Probability with Martingales*, where Chapter 13 covers uniform integrability and Chapter 14 treats UI martingales.[16](https://www.cambridge.org/highereducation/books/probability-with-martingales/B4CFCE0D08930FB46C6E93E775503926/uniform-integrability/2A53700DFF6A472794B6B46190CD4EBC) The concept is also formalized in proof assistants: the Lean mathematical library Mathlib contains both the measure-theoretic and the probabilistic definitions together with the Vitali convergence theorem.[10](https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/UniformIntegrable.html)
## References
1. [Chapter 7: Martingales, §7.4 Uniform Integrability, D. McDonald, University of Waterloo lecture notes](https://www.math.uwaterloo.ca/~dlmcleis/s901/chapt7.pdf)
2. [Uniform Integrability and the Vitali Theorem, Zackienzle blog](https://zackienzle.com/blog/uniform-integrability)
3. [A. Eberle, Stochastic Analysis, §4.3 Uniform Integrability and L¹ Convergence, University of Bonn](https://wt.iam.uni-bonn.de/fileadmin/WT/Inhalt/people/Andreas_Eberle/IntroStoAn1819/StochasticAnalysisComplete.pdf)
4. [A. Dembo, Probability Theory: STAT310/MATH230, Stanford lecture notes](https://adembo.su.domains/stat-310b/lnotes.pdf)
5. [H. Breis, Functional Analysis, Sobolev Spaces and PDEs, Theorem 4.30 (Dunford–Pettis)](https://www.math.toronto.edu/almut/Brezis.pdf)
6. [D. Chafaï, De la Vallée Poussin on Uniform Integrability (2014)](https://djalil.chafai.net/blog/2014/03/09/de-la-vallee-poussin-on-uniform-integrability/)
7. [S. Drapeau, Uniform Integrability, Stochastics lecture notes](https://www.samuel-drapeau.info/SP_Lecture/lecture/03-Integration/036-uniform-integrability/)
8. [N. Forien, A review of probability theory foundations, Université Paris-Dauphine](https://www.ceremade.dauphine.fr/~forien/Nicolas_Forien_files/poly-prerentree-m2.pdf)
9. [Conditional Expectation and Martingales, Rutgers Financial Mathematics lecture notes](https://finmath.rutgers.edu/images/documents/finance/Martingales.pdf)
10. [Mathlib.MeasureTheory.Function.UniformIntegrable, Lean Mathlib documentation](https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/UniformIntegrable.html)
11. [Uniform integrability of exponential martingales and spectral bounds of non-local Feynman-Kac semigroups, arXiv:1105.3020](https://arxiv.org/abs/1105.3020)
12. [The Dunford-Pettis theorem, J. Bell](https://jordanbell.info/LaTeX/mathematics/dunford-pettis/)
13. [R. Rossi, G. Savaré, Tightness, Integral Equicontinuity and Compactness for Evolution Problems in Banach Spaces, Ann. Scuola Norm. Sup. Pisa (2003)](https://www.numdam.org/article/ASNSP_2003_5_2_2_395_0.pdf)
14. [Weak Compactness Criterion in W^{k,1} with an Existence Theorem of Minimizers, ResearchGate (2023)](https://www.researchgate.net/publication/371943814_Weak_Compactness_Criterion_in_Wk_1_with_an_Existence_Theorem_of_Minimizers)
15. [C. De La Vallée Poussin, Sur l'intégrale de Lebesgue, Transactions of the AMS 16(4), 435–501 (1915)](https://cir.nii.ac.jp/crid/1360292620414059776)
16. [D. Williams, Probability with Martingales, Chapter 13: Uniform Integrability, Cambridge University Press](https://www.cambridge.org/highereducation/books/probability-with-martingales/B4CFCE0D08930FB46C6E93E775503926/uniform-integrability/2A53700DFF6A472794B6B46190CD4EBC)
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