Uniform integrability

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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 also known as equi-integrability.^{[1] : Definition 2.23}

Uniform integrability is an extension to the notion of a family of functions being dominated in ${\displaystyle L_{1}}${\displaystyle L_{1}} which is central in dominated convergence. Several textbooks on real analysis and measure theory use the following definition:^[2]

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

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

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

Definition A is rather restrictive for infinite measure spaces. A more general definition^[3] of uniform integrability that works well in general measure spaces was introduced by G. A. Hunt.

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

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

where ${\displaystyle L_{+}^{1}(\mu )=\{g\in L^{1}(\mu ):g\geq 0\}}${\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 result^[4] provides another equivalent notion to Hunt's. This equivalency is sometimes given as definition for uniform integrability.

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

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

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

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

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

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

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

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

whenever ${\displaystyle \int _{A}h,円d\mu <\delta }${\displaystyle \int _{A}h,円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 ${\displaystyle h\equiv 1}${\displaystyle h\equiv 1} in Theorem 2.

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.,^{[5] [6] [7]} that is,

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

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

or alternatively

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

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

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

A tight family with respect to ${\displaystyle \Phi ={\mathfrak {M}}\cap L_{1}(,円u)}${\displaystyle \Phi ={\mathfrak {M}}\cap L_{1}(,円u)} is just said to be tight.

When the measure space ${\displaystyle (X,{\mathfrak {M}},\mu )}${\displaystyle (X,{\mathfrak {M}},\mu )} is a metric space equipped with the Borel ${\displaystyle \sigma }${\displaystyle \sigma } algebra, ${\displaystyle \mu }${\displaystyle \mu } is a regular measure, and ${\displaystyle {\mathcal {K}}}${\displaystyle {\mathcal {K}}} is the collection of all compact subsets of ${\displaystyle X}${\displaystyle X}, the notion of ${\displaystyle {\mathcal {K}}}${\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 ${\displaystyle \sigma }${\displaystyle \sigma }-finite measure spaces, it can be shown that if a family ${\displaystyle \Phi \subset L_{1}(\mu )}${\displaystyle \Phi \subset L_{1}(\mu )} is uniformly integrable, then ${\displaystyle \Phi }${\displaystyle \Phi } is tight. This is capture by the following result which is often used as definition of uniform integrabiliy in the Analysis literature:

Theorem 3: Suppose ${\displaystyle (X,{\mathfrak {M}},\mu )}${\displaystyle (X,{\mathfrak {M}},\mu )} is a ${\displaystyle \sigma }${\displaystyle \sigma } finite measure space. A family ${\displaystyle \Phi \subset L_{1}(\mu )}${\displaystyle \Phi \subset L_{1}(\mu )} is uniformly integrable if and only if

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

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

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 ${\displaystyle (\Omega ,{\mathcal {F}},P)}${\displaystyle (\Omega ,{\mathcal {F}},P)} is a probability space. A class ${\displaystyle {\mathcal {C}}}${\displaystyle {\mathcal {C}}} of random variables is uniformly absolutely continuous with respect to ${\displaystyle P}${\displaystyle P} if for any ${\displaystyle \varepsilon >0}${\displaystyle \varepsilon >0}, there is ${\displaystyle \delta >0}${\displaystyle \delta >0} such that ${\displaystyle E[|X|I_{A}]<\varepsilon }${\displaystyle E[|X|I_{A}]<\varepsilon } whenever ${\displaystyle P(A)<\delta }${\displaystyle P(A)<\delta }.

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]

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

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 ${\displaystyle L^{1}(\mu )}${\displaystyle L^{1}(\mu )}.

• Dunford–Pettis theorem^{[12] [13]} A class^{[clarification needed ]} of random variables ${\displaystyle X_{n}\subset L^{1}(\mu )}${\displaystyle X_{n}\subset L^{1}(\mu )} is uniformly integrable if and only if it is relatively compact for the weak topology ${\displaystyle \sigma (L^{1},L^{\infty })}${\displaystyle \sigma (L^{1},L^{\infty })}.^{[clarification needed ][citation needed ]}
• de la Vallée-Poussin theorem^{[14] [15]} The family ${\displaystyle \{X_{\alpha }\}_{\alpha \in \mathrm {A} }\subset L^{1}(\mu )}${\displaystyle \{X_{\alpha }\}_{\alpha \in \mathrm {A} }\subset L^{1}(\mu )} is uniformly integrable if and only if there exists a non-negative increasing convex function ${\displaystyle G(t)}${\displaystyle G(t)} such that $${\displaystyle \lim _{t\to \infty }{\frac {G(t)}{t}}=\infty {\text{ and }}\sup _{\alpha }\operatorname {E} (G(|X_{\alpha }|))<\infty .}$${\displaystyle \lim _{t\to \infty }{\frac {G(t)}{t}}=\infty {\text{ and }}\sup _{\alpha }\operatorname {E} (G(|X_{\alpha }|))<\infty .}

A family of random variables ${\displaystyle \{X_{i}\}_{i\in I}}${\displaystyle \{X_{i}\}_{i\in I}} is uniformly integrable if and only if^[16] there exists a random variable ${\displaystyle X}${\displaystyle X} such that ${\displaystyle EX<\infty }${\displaystyle EX<\infty } and ${\displaystyle |X_{i}|\leq _{\mathrm {icx} }X}${\displaystyle |X_{i}|\leq _{\mathrm {icx} }X} for all ${\displaystyle i\in I}${\displaystyle i\in I}, where ${\displaystyle \leq _{\mathrm {icx} }}${\displaystyle \leq _{\mathrm {icx} }} denotes the increasing convex stochastic order defined by ${\displaystyle A\leq _{\mathrm {icx} }B}${\displaystyle A\leq _{\mathrm {icx} }B} if ${\displaystyle E\phi (A)\leq E\phi (B)}${\displaystyle E\phi (A)\leq E\phi (B)} for all nondecreasing convex real functions ${\displaystyle \phi }${\displaystyle \phi }.

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

• Shiryaev, A.N. (1995). Probability (2 ed.). New York: Springer-Verlag. pp. 187–188. ISBN 978-0-387-94549-1.
• Diestel, J. and Uhl, J. (1977). Vector measures, Mathematical Surveys 15, American Mathematical Society, Providence, RI ISBN 978-0-8218-1515-1

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