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

which is central in dominated convergence. Several textbooks on real analysis and measure theory use the following definition:

Definition A: be a positive measure space is called uniformly integrable such that

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

Definition H: is called uniformly integrable if and only if

.

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 provides another equivalent notion to Hunt's. This equivalency is sometimes given as definition for uniform integrability.

Theorem 1: is uniformly integrable if and only if

, then uniform integrability is equivalent to either of the following conditions

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

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

-finite, Hunt's definition is equivalent to the following:

Theorem 2: such that

.

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: is said to be tight with respect to if

is simply said to be tight.

is a metric space is a regular measure-tightness discussed above coincides with the well known concept of tightness used in the analysis of regular measures in metric spaces

is tight. This is captured by the following result which is often used as definition of uniform integrability in the analysis literature:

Theorem 3: is uniformly integrable if and only if

1. .

2. is tight.

, condition 3 is redundant (see Theorem 1 above).

In many books in analysis , condition 2 in Theorem 3 is often replaced by another condition called equi-integrability:

Definition: of complex or real valued measurable functions is equi-integrable (or uniformly absolutely continuous Theorem 3 then says that equi-integrability 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 can be chosen to be monotone increasing and convex.

.

Dunford–Pettis theorem has compact closure in the weak topology 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., that is,

of random variables is called uniformly integrable if:

• and

• .

or alternatively

of random variables is called uniformly integrable is the indicator function.

Related corollaries

The following results apply to the probabilistic definition.

• for all n and comparing with definition 1, it is seen that the sequence is not uniformly integrable.

• .

• is a UI.

• : that is, for all ω and n is uniformly integrable.

• ) is uniformly integrable.

Uniform integrability and stochastic ordering

is uniformly integrable if and only if denotes the increasing convex stochastic order.

Relation to convergence of random variables

norm if and only if it converges in measure 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. This is a generalization of Lebesgue's dominated convergence theorem, see Vitali convergence theorem.

Citations

References

• Diestel, J. and Uhl, J. (1977). Vector measures, Mathematical Surveys 15, American Mathematical Society, Providence, RI