Uniform integrability


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

Definition A: Let be a positive measure space. A set is called uniformly integrable if, and to each there corresponds a such that
whenever and

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: Let be a positive measure space. A set is called uniformly integrable if and only if
where.

Since Hunt's definition is equivalent to Definition A when the underlying measure space is finite, 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: If is a finite measure space, then a set is uniformly integrable if and only if
If in addition, then uniform integrability is equivalent to either of the following conditions
1..
2.

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

Theorem 2: Let be a -finite measure space, and be such that almost everywhere. A set is uniformly integrable if and only if, and for any, there exists such that
whenever.

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 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 is a measure space. Let be a collection of sets of finite measure. A family is said to be tight with respect to if
When, is simply said to be tight.

When the measure space is a metric space equipped with the Borel algebra, is a regular measure, and is the collection of all compact subsets of, the notion of -tightness discussed above coincides with the well known concept of tightness used in the analysis of regular measures in metric spaces
For -finite measure spaces, it can be shown that if a family is uniformly integrable, then 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 is a -finite measure space. A family is uniformly integrable if and only if
  1. .
  3. is tight.
When, condition 3 is redundant.

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

Definition: A family of complex or real valued measurable functions is equi-integrable if for any there is such that

Theorem 3 then says that equi-integrability together with boundedness and tightness and 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 Suppose is a finite measure space. The family is uniformly integrable if and only if there exists a function such that and
The function can be chosen to be monotone increasing and convex.

Uniform integrability gives a characterization of weak compactness in.
Dunford–Pettis theorem Suppose is a -finite measure. A family has compact closure in the weak topology if and only if 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,
1. A class of random variables is called uniformly integrable if:
  • There exists a finite such that, for every in, and
  • For every there exists such that, for every measurable such that and every in, .
or alternatively
2. A class of random variables is called uniformly integrable if for every there exists such that, where is the indicator function .

Related corollaries

The following results apply to the probabilistic definition.
  • Definition 1 could be rewritten by taking the limits as
  • A non-UI sequence. Let, and define Clearly, and indeed for all n. However, 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 norm of all s are 1 i.e., bounded. But the second clause does not hold as given any positive, there is an interval with measure less than and for all.
  • If is a UI random variable, by splitting and bounding each of the two, it can be seen that a uniformly integrable random variable is always bounded in.
  • If any sequence of random variables is dominated by an integrable, non-negative : that is, for all ω and n, then the class of random variables is uniformly integrable.
  • A class of random variables bounded in is uniformly integrable.

    Uniform integrability and stochastic ordering

A family of random variables is uniformly integrable if and only if
there exists a random variable
such that and
for all, where
denotes the increasing convex stochastic order defined by
if for all nondecreasing convex real functions.

Relation to convergence of random variables

A sequence converges to in the norm if and only if it converges in measure to 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.