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 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 measures 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 . For finite measure spaces the following result follows from Definition H: Theorem 1: If is a (positive) finite measure space, then a set is uniformly integrable if and only if Many textbooks in probability present Theorem 1 as the definition of uniform integrability in Probability spaces. When the space is -finite, Definition H yields the following equivalency: Theorem 2: Let be a -finite measure space, and be such that almost surely. A set is uniformly integrable if and only if , and for any , there exits such that whenever . In particular, the equivalence of Definitions A and H for finite measures follows immediately from Theorem 2; for this case, the statement in Definition A is obtained by taking in Theorem 2. 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., 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 (UI) if for every there exists such that , where is the indicator function .