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

Measure-theoretic definition

Uniform integrability is an extension to the notion of a family of functions being dominated in L^1 which is central in dominated convergence. Several textbooks on real analysis and measure theory use the following definition:[1]

Definition A: Let (X,\mathfrak{M}, \mu) be a positive measure space. A set \Phi\subset L^1(\mu) is called uniformly integrable if \sup_{f\in\Phi}\|f\|_{L^1(\mu)}<\infty, and to each \varepsilon>0 there corresponds a \delta>0 such that

\int_E |f| \, d\mu < \varepsilon

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

Definition A is rather restrictive for infinite measure spaces. A more general definition[2] of uniform integrability that works well in general measure spaces was introduced by G. A. Hunt.

Definition H: Let (X,\mathfrak{M},\mu) be a positive measure space. A set \Phi\subset L^1(\mu) is called uniformly integrable if and only if

\inf_{g\in L^1_+(\mu)}\sup_{f\in\Phi}\int_{\{|f|>g\}}|f|\, d\mu=0

where L^1_+(\mu)=\{g\in L^1(\mu): g\geq0\}.

Since Hunt's definition is equivalent to Definition A when the underlying measure space is finite (see Theorem 2 below), Definition H is widely adopted in Mathematics.

The following result[3] provides another equivalent notion to Hunt's. This equivalency is sometimes given as definition for uniform integrability.

Theorem 1: If (X,\mathfrak{M},\mu) is a (positive) finite measure space, then a set \Phi\subset L^1(\mu) is uniformly integrable if and only if

\inf_{g\in L^1_+(\mu)}\sup_{f\in\Phi}\int (|f|- g)^+ \, d\mu=0

If in addition \mu(X)<\infty, then uniform integrability is equivalent to either of the following conditions

1. \inf_{a>0}\sup_{f\in \Phi}\int(|f|-a)_+\,d\mu =0.

2. \inf_{a>0}\sup_{f\in \Phi}\int_{\{|f|>a\}}|f|\,d\mu=0

When the underlying space (X,\mathfrak{M},\mu) is \sigma-finite, Hunt's definition is equivalent to the following:

Theorem 2: Let (X,\mathfrak{M},\mu) be a \sigma-finite measure space, and h\in L^1(\mu) be such that h>0 almost everywhere. A set \Phi\subset L^1(\mu) is uniformly integrable if and only if \sup_{f\in\Phi}\|f\|_{L^1(\mu)}<\infty, and for any \varepsilon>0, there exists \delta>0 such that

\sup_{f\in\Phi}\int_A|f|\, d\mu <\varepsilon

whenever \int_A h\,d\mu <\delta.

A consequence of Theorems 1 and 2 is that equivalence of Definitions A and H for finite measures follows. Indeed, the statement in Definition A is obtained by taking h\equiv1 in Theorem 2.

Tightness, boundedness, equi-integrability and uniform integrability

Another concept associated with uniform integrability is that of tightness. In this article tightness is taken in a more general setting.

Definition: Suppose (X,\mathfrak{M},\mu) is a measure space. Let \mathcal{K}\subset\mathfrak{M} be a collection of sets of finite measure. A family \Phi\subset L^1(\mu) is said to be tight with respect to \mathcal{K} if

\inf_{K\in\mathcal{K}}\sup_{f\in\Phi}\int_{X\setminus K}|f|\,d\mu=0

When \mathcal{K}=\mathfrak{M}\cap L^1(\mu), \Phi is simply said to be tight.

When the measure space (X,\mathfrak{M},\mu) is a metric space equipped with the Borel \sigma algebra, \mu is a regular measure, and \mathcal{K} is the collection of all compact subsets of X, the notion of \mathcal{K}-tightness discussed above coincides with the well known concept of tightness used in the analysis of regular measures in metric spaces

For \sigma-finite measure spaces, it can be shown that if a family \Phi\subset L^1(\mu) is uniformly integrable, then \Phi is tight. This is captured by the following result which is often used as definition of uniform integrability in the analysis literature:

Theorem 3: Suppose (X,\mathfrak{M},\mu) is a \sigma-finite measure space. A family \Phi\subset L^1(\mu) is uniformly integrable if and only if

  1. \sup_{f\in\Phi}\|f\|_1<\infty.
  2. \inf_{a>0}\sup_{f\in \Phi}\int_{\{|f|>a\}}|f|\,d\mu=0
  3. \Phi is tight.

When \mu(X)<\infty, condition 3 is redundant (see Theorem 1 above).

In many books in analysis [4][5][6][7], condition 2 in Theorem 3 is often replaced by another condition called equi-integrability:

Definition: A family \mathcal{C} of complex or real valued measurable functions is equi-integrable (or uniformly absolutely continuous with respect to a measure \mu) if for any \varepsilon>0 there is \delta>0 such that

\sup_{f\in\mathcal{C}}\int_A|f|\,d\mu<\varepsilon \qquad\text{whenever}\qquad \mu(A)<\delta

Theorem 3 then says that equi-integrability together with L^1 boundedness and tightness (conditions (1) and (3) in Theorem 3) is equivalent to uniform integrability.

Relevant theorems

The following theorems describe very useful criteria for uniform integrability which have many applications in Analysis and Probability.

de la Vallée-Poussin theorem[8][9] Suppose (X,\mathfrak{M},\mu) is a finite measure space. The family \mathcal{F} \subset L^1(\mu) is uniformly integrable if and only if there exists a function G:[0,\infty)\rightarrow[0,\infty) such that \lim_{t \to \infty} \frac{G(t)} t = \infty and 
\sup_{f\in\mathcal{F}} \int_X G(|f|)\,d\mu < \infty.

The function G can be chosen to be monotone increasing and convex.

Uniform integrability gives a characterization of weak compactness in L^1.

Dunford–Pettis theorem[10][11] Suppose (X,\mathfrak{M},\mu) is a \sigma-finite measure. A family \mathcal{F}\subset L^1(\mu) has compact closure in the weak topology \sigma(L^1,L^\infty) if and only if \mathcal{F} is uniformly integrable.

Probability definition

In probability theory, Definition A or the statement of Theorem 1 are often presented as definitions of uniform integrability using the notation expectation of random variables.,[12][13][14] that is,

1. A class \mathcal{C} of random variables is called uniformly integrable if:

  • There exists a finite M such that, for every X in \mathcal{C}, \operatorname E(|X|)\leq M and
  • For every \varepsilon > 0 there exists \delta > 0 such that, for every measurable A such that P(A)\leq \delta and every X in \mathcal{C}, \operatorname E(|X|I_A)\leq\varepsilon.

or alternatively

2. A class \mathcal{C} of random variables is called uniformly integrable (UI) if for every \varepsilon > 0 there exists K\in[0,\infty) such that \operatorname E(|X|I_{|X|\geq K})\le\varepsilon\ \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}.

The following results apply to the probabilistic definition.[15]

  • Definition 1 could be rewritten by taking the limits as
\lim_{K \to \infty} \sup_{X \in \mathcal{C}} \operatorname  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 \operatorname E(|X_n|)=1\ , for all n. However,

\operatorname E(|X_n| I_{\{|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.

  • 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_ns 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
\operatorname E(|X|) = \operatorname E(|X| I_{\{|X| \geq K \}})+\operatorname E(|X| I_{\{|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)| \le Y(\omega),\ Y(\omega)\ge 0,\ \operatorname 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.

Uniform integrability and stochastic ordering

A family of random variables \{X_i\}_{i \in I} is uniformly integrable if and only if[16] there exists a random variable X such that E X < \infty and |X_i| \le_\mathrm{icx} X for all i \in I, where \le_\mathrm{icx} denotes the increasing convex stochastic order defined by A \le_\mathrm{icx} B if E \phi(A) \le E \phi(B) for all nondecreasing convex real functions \phi.

Relation to convergence of random variables

Main article: 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.[17] This is a generalization of Lebesgue's dominated convergence theorem, see Vitali convergence theorem.

Citations

  1. ^ Royden, H.L. & Fitzpatrick, P.M. (2010). Real Analysis. 4 ed. Boston: Prentice Hall. p. 93. ISBN 978-0-13-143747-0.
  2. ^ Hunt, G. A. (1966). Martingales et Processus de Markov. Paris: Dunod. p. 33.
  3. ^ Klenke, A. (2008). Probability Theory: A Comprehensive Course. Berlin: Springer Verlag. pp. 134–137. ISBN 978-1-84800-047-6.
  4. ^ Fonseca, Irene & Leoni, Giovanni (2007). Modern Methods in the Calculus of Variations: Lp Spaces. New York, NY: Springer New York Springer e-books. ISBN 978-0387690063.
  5. ^ Benedetto, J. J. (1976). Real Variable and Integration. Stuttgart: B. G. Teubner. p. 89. ISBN 3-519-02209-5.
  6. ^ Burrill, C. W. (1972). Measure, Integration, and Probability. McGraw-Hill. p. 180. ISBN 0-07-009223-0.
  7. ^ Bass, Richard F. (2011). Stochastic Processes. Cambridge: Cambridge University Press. pp. 356–357. ISBN 978-1-107-00800-7.
  8. ^ Meyer, P.A. (1966). Probability and Potentials, Blaisdell Publishing Co, N. Y. (p.19, Theorem T22).
  9. ^ De La Vallée Poussin, C. (1915). "Sur L'Integrale de Lebesgue". Transactions of the American Mathematical Society. 16 (4): 435–501. doi:10.2307/1988879. hdl:10338.dmlcz/127627. JSTOR 1988879
  10. ^ Dunford, Nelson (1938). "Uniformity in linear spaces". Transactions of the American Mathematical Society. 44 (2): 305–356. doi:10.1090/S0002-9947-1938-1501971-X. ISSN 0002-9947
  11. ^ Dunford, Nelson (1939). "A mean ergodic theorem". Duke Mathematical Journal. 5 (3): 635–646. doi:10.1215/S0012-7094-39-00552-1. ISSN 0012-7094
  12. ^ Williams, David (1997). Probability with Martingales. Repr. ed. Cambridge: Cambridge Univ. Press. pp. 126–132. ISBN 978-0-521-40605-5.
  13. ^ Gut, Allan (2005). Probability: A Graduate Course. Springer. pp. 214–218. ISBN 0-387-22833-0.
  14. ^ Bass, Richard F. (2011). Stochastic Processes. Cambridge: Cambridge University Press. pp. 356–357. ISBN 978-1-107-00800-7.
  15. ^ Gut 2005, pp. 215–216.
  16. ^ Leskelä, L. & Vihola, M. (2013). "Stochastic order characterization of uniform integrability and tightness". Statistics and Probability Letters. 83 (1): 382–389. arXiv:1106.0607. doi:10.1016/j.spl.2012.09.023
  17. ^ Bogachev, Vladimir I. (2007). "The spaces Lp and spaces of measures". Measure Theory Volume I. Berlin Heidelberg: Springer-Verlag. p. 268. doi:10.1007/978-3-540-34514-5_4. ISBN 978-3-540-34513-8.

References

  • Shiryaev, A.N. (1995). Probability. 2 ed. New York: Springer-Verlag. pp. 187–188. ISBN 978-0-387-94549-1.
  • Diestel, J. and Uhl, J. (1977). Vector measures, Mathematical Surveys 15, American Mathematical Society, Providence, RI ISBN 978-0-8218-1515-1