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 ${\displaystyle L^{1}}$ which is central in dominated convergence. Several textbooks on real analysis and measure theory use the following definition:¹

Definition A: Let ${\displaystyle (X,{\mathfrak {M}},\mu )}$ be a positive measure space. A set ${\displaystyle \Phi \subset L^{1}(\mu )}$ is called uniformly integrable if ${\displaystyle \sup _{f\in \Phi }\|f\|_{L^{1}(\mu )}<\infty }$, and to each ${\displaystyle \varepsilon >0}$ there corresponds a ${\displaystyle \delta >0}$ such that

${\displaystyle \int _{E}|f|\,d\mu <\varepsilon }$

whenever ${\displaystyle f\in \Phi }$ and ${\displaystyle \mu (E)<\delta .}$

Definition A is rather restrictive for infinite measure spaces. A more general definition² of uniform integrability that works well in general measure spaces was introduced by G. A. Hunt.

Definition H: Let ${\displaystyle (X,{\mathfrak {M}},\mu )}$ be a positive measure space. A set ${\displaystyle \Phi \subset L^{1}(\mu )}$ is called uniformly integrable if and only if

${\displaystyle \inf _{g\in L_{+}^{1}(\mu )}\sup _{f\in \Phi }\int _{\{|f|>g\}}|f|\,d\mu =0}$

where ${\displaystyle L_{+}^{1}(\mu )=\{g\in L^{1}(\mu ):g\geq 0\}}$.

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 result³ provides another equivalent notion to Hunt's. This equivalency is sometimes given as definition for uniform integrability.

Theorem 1: If ${\displaystyle (X,{\mathfrak {M}},\mu )}$ is a (positive) finite measure space, then a set ${\displaystyle \Phi \subset L^{1}(\mu )}$ is uniformly integrable if and only if

${\displaystyle \inf _{g\in L_{+}^{1}(\mu )}\sup _{f\in \Phi }\int (|f|-g)^{+}\,d\mu =0}$

If in addition ${\displaystyle \mu (X)<\infty }$, then uniform integrability is equivalent to either of the following conditions

1. ${\displaystyle \inf _{a>0}\sup _{f\in \Phi }\int (|f|-a)_{+}\,d\mu =0}$.

2. ${\displaystyle \inf _{a>0}\sup _{f\in \Phi }\int _{\{|f|>a\}}|f|\,d\mu =0}$

When the underlying space ${\displaystyle (X,{\mathfrak {M}},\mu )}$ is ${\displaystyle \sigma }$-finite, Hunt's definition is equivalent to the following:

Theorem 2: Let ${\displaystyle (X,{\mathfrak {M}},\mu )}$ be a ${\displaystyle \sigma }$-finite measure space, and ${\displaystyle h\in L^{1}(\mu )}$ be such that ${\displaystyle h>0}$ almost everywhere. A set ${\displaystyle \Phi \subset L^{1}(\mu )}$ is uniformly integrable if and only if ${\displaystyle \sup _{f\in \Phi }\|f\|_{L^{1}(\mu )}<\infty }$, and for any ${\displaystyle \varepsilon >0}$, there exists ${\displaystyle \delta >0}$ such that

${\displaystyle \sup _{f\in \Phi }\int _{A}|f|\,d\mu <\varepsilon }$

whenever ${\displaystyle \int _{A}h\,d\mu <\delta }$.

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 ${\displaystyle h\equiv 1}$ in Theorem 2.

Another concept associated with uniform integrability is that of tightness. In this article tightness is taken in a more general setting.

Definition: Suppose ${\displaystyle (X,{\mathfrak {M}},\mu )}$ is a measure space. Let ${\displaystyle {\mathcal {K}}\subset {\mathfrak {M}}}$ be a collection of sets of finite measure. A family ${\displaystyle \Phi \subset L^{1}(\mu )}$ is said to be tight with respect to ${\displaystyle {\mathcal {K}}}$ if

${\displaystyle \inf _{K\in {\mathcal {K}}}\sup _{f\in \Phi }\int _{X\setminus K}|f|\,d\mu =0}$

When ${\displaystyle {\mathcal {K}}={\mathfrak {M}}\cap L^{1}(\mu )}$, ${\displaystyle \Phi }$ is simply said to be tight.

When the measure space ${\displaystyle (X,{\mathfrak {M}},\mu )}$ is a metric space equipped with the Borel ${\displaystyle \sigma }$ algebra, ${\displaystyle \mu }$ is a regular measure, and ${\displaystyle {\mathcal {K}}}$ is the collection of all compact subsets of ${\displaystyle X}$, the notion of ${\displaystyle {\mathcal {K}}}$-tightness discussed above coincides with the well known concept of tightness used in the analysis of regular measures in metric spaces

For ${\displaystyle \sigma }$-finite measure spaces, it can be shown that if a family ${\displaystyle \Phi \subset L^{1}(\mu )}$ is uniformly integrable, then ${\displaystyle \Phi }$ 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 ${\displaystyle (X,{\mathfrak {M}},\mu )}$ is a ${\displaystyle \sigma }$-finite measure space. A family ${\displaystyle \Phi \subset L^{1}(\mu )}$ is uniformly integrable if and only if

1. ${\displaystyle \sup _{f\in \Phi }\|f\|_{1}<\infty }$.
2. ${\displaystyle \inf _{a>0}\sup _{f\in \Phi }\int _{\{|f|>a\}}|f|\,d\mu =0}$
3. ${\displaystyle \Phi }$ is tight.

When ${\displaystyle \mu (X)<\infty }$, condition 3 is redundant (see Theorem 1 above).

In many books in analysis ⁴⁵⁶⁷, condition 2 in Theorem 3 is often replaced by another condition called equi-integrability:

Definition: A family ${\displaystyle {\mathcal {C}}}$ of complex or real valued measurable functions is equi-integrable (or uniformly absolutely continuous with respect to a measure ${\displaystyle \mu }$) if for any ${\displaystyle \varepsilon >0}$ there is ${\displaystyle \delta >0}$ such that $${\displaystyle \sup _{f\in {\mathcal {C}}}\int _{A}|f|\,d\mu <\varepsilon \qquad {\text{whenever}}\qquad \mu (A)<\delta }$$

Theorem 3 then says that equi-integrability together with ${\displaystyle L^{1}}$ boundedness and tightness (conditions (1) and (3) in Theorem 3) is equivalent to uniform integrability.

The following theorems describe very useful criteria for uniform integrability which have many applications in Analysis and Probability.

de la Vallée-Poussin theorem⁸⁹

Suppose ${\displaystyle (X,{\mathfrak {M}},\mu )}$ is a finite measure space. The family ${\displaystyle {\mathcal {F}}\subset L^{1}(\mu )}$ is uniformly integrable if and only if there exists a function ${\displaystyle G:[0,\infty )\rightarrow [0,\infty )}$ such that ${\displaystyle \lim _{t\to \infty }{\frac {G(t)}{t}}=\infty }$ and $${\displaystyle \sup _{f\in {\mathcal {F}}}\int _{X}G(|f|)\,d\mu <\infty .}$$ The function ${\displaystyle G}$ can be chosen to be monotone increasing and convex.

Uniform integrability gives a characterization of weak compactness in ${\displaystyle L^{1}}$.

Dunford–Pettis theorem¹⁰¹¹

Suppose ${\displaystyle (X,{\mathfrak {M}},\mu )}$ is a ${\displaystyle \sigma }$-finite measure. A family ${\displaystyle {\mathcal {F}}\subset L^{1}(\mu )}$ has compact closure in the weak topology ${\displaystyle \sigma (L^{1},L^{\infty })}$ if and only if ${\displaystyle {\mathcal {F}}}$ is uniformly integrable.

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.,¹²¹³¹⁴ that is,

1. A class ${\displaystyle {\mathcal {C}}}$ of random variables is called uniformly integrable if:

• There exists a finite ${\displaystyle M}$ such that, for every ${\displaystyle X}$ in ${\displaystyle {\mathcal {C}}}$, ${\displaystyle \operatorname {E} (|X|)\leq M}$ and
• For every ${\displaystyle \varepsilon >0}$ there exists ${\displaystyle \delta >0}$ such that, for every measurable ${\displaystyle A}$ such that ${\displaystyle P(A)\leq \delta }$ and every ${\displaystyle X}$ in ${\displaystyle {\mathcal {C}}}$, ${\displaystyle \operatorname {E} (|X|I_{A})\leq \varepsilon }$.

or alternatively

2. A class ${\displaystyle {\mathcal {C}}}$ of random variables is called uniformly integrable (UI) if for every ${\displaystyle \varepsilon >0}$ there exists ${\displaystyle K\in [0,\infty )}$ such that ${\displaystyle \operatorname {E} (|X|I_{|X|\geq K})\leq \varepsilon \ {\text{ for all }}X\in {\mathcal {C}}}$, where ${\displaystyle I_{|X|\geq K}}$ is the indicator function ${\displaystyle I_{|X|\geq K}={\begin{cases}1&{\text{if }}|X|\geq K,\\0&{\text{if }}|X|<K.\end{cases}}}$.

The following results apply to the probabilistic definition.¹⁵

• Definition 1 could be rewritten by taking the limits as $${\displaystyle \lim _{K\to \infty }\sup _{X\in {\mathcal {C}}}\operatorname {E} (|X|\,I_{|X|\geq K})=0.}$$
• A non-UI sequence. Let ${\displaystyle \Omega =[0,1]\subset \mathbb {R} }$, and define $${\displaystyle X_{n}(\omega )={\begin{cases}n,&\omega \in (0,1/n),\\0,&{\text{otherwise.}}\end{cases}}}$$ Clearly ${\displaystyle X_{n}\in L^{1}}$, and indeed ${\displaystyle \operatorname {E} (|X_{n}|)=1\ ,}$ for all n. However, $${\displaystyle \operatorname {E} (|X_{n}|I_{\{|X_{n}|\geq K\}})=1\ {\text{ for all }}n\geq K,}$$ 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 ${\displaystyle L^{1}}$ norm of all ${\displaystyle X_{n}}$s are 1 i.e., bounded. But the second clause does not hold as given any ${\displaystyle \delta }$ positive, there is an interval ${\displaystyle (0,1/n)}$ with measure less than ${\displaystyle \delta }$ and ${\displaystyle E[|X_{m}|:(0,1/n)]=1}$ for all ${\displaystyle m\geq n}$.
• If ${\displaystyle X}$ is a UI random variable, by splitting $${\displaystyle \operatorname {E} (|X|)=\operatorname {E} (|X|I_{\{|X|\geq K\}})+\operatorname {E} (|X|I_{\{|X|<K\}})}$$ and bounding each of the two, it can be seen that a uniformly integrable random variable is always bounded in ${\displaystyle L^{1}}$.
• If any sequence of random variables ${\displaystyle X_{n}}$ is dominated by an integrable, non-negative ${\displaystyle Y}$: that is, for all ω and n, $${\displaystyle |X_{n}(\omega )|\leq Y(\omega ),\ Y(\omega )\geq 0,\ \operatorname {E} (Y)<\infty ,}$$ then the class ${\displaystyle {\mathcal {C}}}$ of random variables ${\displaystyle \{X_{n}\}}$ is uniformly integrable.
• A class of random variables bounded in ${\displaystyle L^{p}}$ (${\displaystyle p>1}$) is uniformly integrable.

A family of random variables ${\displaystyle \{X_{i}\}_{i\in I}}$ is uniformly integrable if and only if¹⁶ there exists a random variable ${\displaystyle X}$ such that ${\displaystyle EX<\infty }$ and ${\displaystyle |X_{i}|\leq _{\mathrm {icx} }X}$ for all ${\displaystyle i\in I}$, where ${\displaystyle \leq _{\mathrm {icx} }}$ denotes the increasing convex stochastic order defined by ${\displaystyle A\leq _{\mathrm {icx} }B}$ if ${\displaystyle E\phi (A)\leq E\phi (B)}$ for all nondecreasing convex real functions ${\displaystyle \phi }$.

A sequence ${\displaystyle \{X_{n}\}}$ converges to ${\displaystyle X}$ in the ${\displaystyle L^{1}}$ norm if and only if it converges in measure to ${\displaystyle X}$ 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.¹⁷ This is a generalization of Lebesgue's dominated convergence theorem, see Vitali convergence theorem.