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 L 1 {\displaystyle L_{1}} which is central in dominated convergence. Several textbooks on real analysis and measure theory use the following definition:
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.
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.
When the underlying space ( X , M , μ ) {\displaystyle (X,{\mathfrak {M}},\mu )} is σ {\displaystyle \sigma } -finite, Hunt's definition is equivalent to the following:
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 ≡ 1 {\displaystyle h\equiv 1} 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 C {\displaystyle {\mathcal {C}}} of random variables is called uniformly integrable if:
or alternatively
2. A class C {\displaystyle {\mathcal {C}}} of random variables is called uniformly integrable (UI) if for every ε > 0 {\displaystyle \varepsilon >0} there exists K ∈ [ 0 , ∞ ) {\displaystyle K\in [0,\infty )} such that E ⁡ ( | X | I | X | ≥ K ) ≤ ε for all X ∈ C {\displaystyle \operatorname {E} (|X|I_{|X|\geq K})\leq \varepsilon \ {\text{ for all }}X\in {\mathcal {C}}} , where I | X | ≥ K {\displaystyle I_{|X|\geq K}} is the indicator function I | X | ≥ K = { 1 if | X | ≥ K , 0 if | X | < K . {\displaystyle I_{|X|\geq K}={\begin{cases}1&{\text{if }}|X|\geq K,\\0&{\text{if }}|X|<K.\end{cases}}} .
Another concept associated with uniform integrability is that of tightness. In this article tightness is taken in a more general setting.
When the measure space ( X , M , μ ) {\displaystyle (X,{\mathfrak {M}},\mu )} is a metric space equipped with the Borel σ {\displaystyle \sigma } algebra, μ {\displaystyle \mu } is a regular measure, and K {\displaystyle {\mathcal {K}}} is the collection of all compact subsets of X {\displaystyle X} , the notion of K {\displaystyle {\mathcal {K}}} -tightness discussed above coincides with the well known concept of tightness used in the analysis of regular measures in metric spaces
For σ {\displaystyle \sigma } -finite measure spaces, it can be shown that if a family Φ ⊂ L 1 ( μ ) {\displaystyle \Phi \subset L_{1}(\mu )} is uniformly integrable, then Φ {\displaystyle \Phi } is tight. This is capture by the following result which is often used as definition of uniform integrabiliy in the Analysis literature:
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
It is equivalent to uniform integrability if the measure is finite and has no atoms.
The term "uniform absolute continuity" is not standard, but is used by some authors.
The following results apply to the probabilistic definition.
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 L 1 ( μ ) {\displaystyle L^{1}(\mu )} .
A family of random variables { X i } i ∈ I {\displaystyle \{X_{i}\}_{i\in I}} is uniformly integrable if and only if there exists a random variable X {\displaystyle X} such that E X < ∞ {\displaystyle EX<\infty } and | X i | ≤ i c x X {\displaystyle |X_{i}|\leq _{\mathrm {icx} }X} for all i ∈ I {\displaystyle i\in I} , where ≤ i c x {\displaystyle \leq _{\mathrm {icx} }} denotes the increasing convex stochastic order defined by A ≤ i c x B {\displaystyle A\leq _{\mathrm {icx} }B} if E ϕ ( A ) ≤ E ϕ ( B ) {\displaystyle E\phi (A)\leq E\phi (B)} for all nondecreasing convex real functions ϕ {\displaystyle \phi } .
A sequence { X n } {\displaystyle \{X_{n}\}} converges to X {\displaystyle X} in the L 1 {\displaystyle L_{1}} norm if and only if it converges in measure to X {\displaystyle 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. This is a generalization of Lebesgue's dominated convergence theorem, see Vitali convergence theorem.