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.
Uniform integrability is an extension to the notion of a family of functions being dominated in
Definition A is rather restrictive for infinite measure spaces.
The following result[4] provides another equivalent notion to Hunt's.
This equivalency is sometimes given as definition for uniform integrability.
is a (positive) finite measure space, then a set
, then uniform integrability is equivalent to either of the following conditions 1.
-finite, Hunt's definition is equivalent to the following: Theorem 2: Let
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 the theory of probability, Definition A or the statement of Theorem 1 are often presented as definitions of uniform integrability using the notation expectation of random variables.,[5][6][7] that is, 1.
of random variables is called uniformly integrable if: or alternatively 2.
of random variables is called uniformly integrable (UI) if for every
Another concept associated with uniform integrability is that of tightness.
In this article tightness is taken in a more general setting.
Definition: Suppose measurable space
is a metric space equipped with the Borel
-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
This is capture by the following result which is often used as definition of uniform integrabiliy in the Analysis literature: Theorem 3: Suppose
There is another notion of uniformity, slightly different than uniform integrability, which also has many applications in probability and measure theory, and which does not require random variables to have a finite integral[8] Definition: Suppose
of random variables is uniformly absolutely continuous with respect to
It is equivalent to uniform integrability if the measure is finite and has no atoms.
The term "uniform absolute continuity" is not standard,[citation needed] but is used by some authors.
[9][10] The following results apply to the probabilistic definition.
[11] In the following we use the probabilistic framework, but regardless of the finiteness of the measure, by adding the boundedness condition on the chosen subset of
is uniformly integrable if and only if[16] there exists a random variable
denotes the increasing convex stochastic order defined by
for all nondecreasing convex real functions
In probability terms, a sequence of random variables converging in probability also converge in the mean if and only if they are uniformly integrable.
