In mathematics, uniform integrability is an important concept in
real analysis
In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include co ...
,
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics ...
and
measure theory
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude (mathematics), magnitude, mass, and probability of events. These seemingl ...
, and plays a vital role in the theory of
martingales.
Measure-theoretic definition
Uniform integrability is an extension to the notion of a family of functions being dominated in
which is central in
dominated convergence
Domination or dominant may refer to:
Society
* World domination, structure where one dominant power governs the planet
* Colonialism in which one group (usually a nation) invades another region for material gain or to eliminate competition
* Cha ...
.
Several textbooks on real analysis and measure theory use the following definition:
Definition A: Let be a positive measure space
A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that ...
. 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 .
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.
Theorem 1: If is a (positive) finite measure space, then a set is uniformly integrable if and only if
:
If in addition , then uniform integrability is equivalent to either of the following conditions
1. .
2.
When the underlying space
is
-finite, Hunt's definition is equivalent to the following:
Theorem 2: Let be a -finite measure space, and be such that almost everywhere. A set is uniformly integrable if and only if , and for any , there exits such that
:
whenever .
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 Theorem 2.
Probability definition
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 variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...
s is called uniformly integrable (UI) if for every
there exists
whenever
P(A)<\delta.
It is equivalent to uniform integrability if the measure is finite and has no atoms.
The term "uniform
absolute continuity
In calculus and real analysis, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between ...
" is not standard, but is used by some authors.
Related corollaries
The following results apply to the probabilistic definition.
* Definition 1 could be rewritten by taking the limits as
\lim_ \sup_ \operatorname E(, X, \,I_)=0.
* A non-UI sequence. Let
\Omega = ,1\subset \mathbb, and define
X_n(\omega) = \begin
n, & \omega\in (0,1/n), \\
0 , & \text \end Clearly
X_n\in L^1, and indeed
\operatorname E(, X_n, )=1\ , for all ''n''. However,
\operatorname E(, X_n, I_)= 1\ \text n \ge 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
L^1 norm of all
X_ns are 1 i.e., bounded. But the second clause does not hold as given any
\delta positive, there is an interval
(0, 1/n) with measure less than
\delta and
E for all m \ge n .
* If X is a UI random variable, by splitting \operatorname E(, X, ) = \operatorname E(, X, I_)+\operatorname E(, X, I_) and bounding each of the two, it can be seen that a uniformly integrable random variable is always bounded in L^1.
* If any sequence of random variables X_n is dominated by an integrable, non-negative Y: that is, for all ''ω'' and ''n'', , X_n(\omega), \le Y(\omega),\ Y(\omega)\ge 0,\ \operatorname E(Y) < \infty, then the class \mathcal of random variables \ is uniformly integrable.
* A class of random variables bounded in L^p (p > 1) is uniformly integrable.
Relevant theorems
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(\mu).
* Nelson Dunford, Dunford– Pettis theoremA class of random variables X_n \subset L^1(\mu) is uniformly integrable if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
it is relatively compact
In mathematics, a relatively compact subspace (or relatively compact subset, or precompact subset) of a topological space is a subset whose closure is compact.
Properties
Every subset of a compact topological space is relatively compact (sinc ...
for the weak topology
In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a ...
\sigma(L^1,L^\infty).
* de la Vallée-Poussin theoremThe family \_ \subset L^1(\mu) is uniformly integrable if and only if there exists a non-negative increasing convex function G(t) such that \lim_ \frac t = \infty \text \sup_\alpha \operatorname E(G(, X_, )) < \infty.
Uniform integrability and stochastic ordering
A family of random variables \_ is uniformly integrable if and only if
there exists a random variable
X such that E X < \infty and
, X_i, \le_\mathrm X for all i \in I, where
\le_\mathrm denotes the increasing convex stochastic order defined by
A \le_\mathrm B if E \phi(A) \le E \phi(B) for all nondecreasing convex real functions \phi.
Relation to convergence of random variables
A sequence \ converges to X in the L_1 norm if and only if it converges in measure to 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
In measure theory, Lebesgue's dominated convergence theorem gives a mild sufficient condition under which limits and integrals of a sequence of functions can be interchanged. More technically it says that if a sequence of functions is bounded i ...
, see Vitali convergence theorem
In real analysis and measure theory, the Vitali convergence theorem, named after the Italian mathematician Giuseppe Vitali, is a generalization of the better-known dominated convergence theorem of Henri Lebesgue. It is a characterization of th ...
.
Citations
References
*
* Diestel, J. and Uhl, J. (1977). ''Vector measures'', Mathematical Surveys 15, American Mathematical Society, Providence, RI {{isbn, 978-0-8218-1515-1
Martingale theory>X, I_A\varepsilon
whenever
P(A)<\delta.
It is equivalent to uniform integrability if the measure is finite and has no atoms.
The term "uniform
absolute continuity
In calculus and real analysis, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between ...
" is not standard, but is used by some authors.
Related corollaries
The following results apply to the probabilistic definition.
* Definition 1 could be rewritten by taking the limits as
\lim_ \sup_ \operatorname E(, X, \,I_)=0.
* A non-UI sequence. Let
\Omega = ,1\subset \mathbb, and define
X_n(\omega) = \begin
n, & \omega\in (0,1/n), \\
0 , & \text \end Clearly
X_n\in L^1, and indeed
\operatorname E(, X_n, )=1\ , for all ''n''. However,
\operatorname E(, X_n, I_)= 1\ \text n \ge 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
L^1 norm of all
X_ns are 1 i.e., bounded. But the second clause does not hold as given any
\delta positive, there is an interval
(0, 1/n) with measure less than
\delta and
E X_m, : (0, 1/n)=1 for all
m \ge n .
* If
X is a UI random variable, by splitting
\operatorname E(, X, ) = \operatorname E(, X, I_)+\operatorname E(, X, I_) and bounding each of the two, it can be seen that a uniformly integrable random variable is always bounded in
L^1.
* If any sequence of random variables
X_n is dominated by an integrable, non-negative
Y: that is, for all ''ω'' and ''n'',
, X_n(\omega), \le Y(\omega),\ Y(\omega)\ge 0,\ \operatorname E(Y) < \infty, then the class
\mathcal of random variables
\ is uniformly integrable.
* A class of random variables bounded in
L^p (
p > 1) is uniformly integrable.
Relevant theorems
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(\mu).
*
Nelson Dunford, Dunford–
Pettis theoremA class of random variables
X_n \subset L^1(\mu) is uniformly integrable
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
it is
relatively compact
In mathematics, a relatively compact subspace (or relatively compact subset, or precompact subset) of a topological space is a subset whose closure is compact.
Properties
Every subset of a compact topological space is relatively compact (sinc ...
for the
weak topology
In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a ...
\sigma(L^1,L^\infty).
*
de la Vallée-Poussin theoremThe family
\_ \subset L^1(\mu) is uniformly integrable if and only if there exists a non-negative increasing convex function
G(t) such that
\lim_ \frac t = \infty \text \sup_\alpha \operatorname E(G(, X_, )) < \infty.
Uniform integrability and stochastic ordering
A family of random variables
\_ is uniformly integrable if and only if
there exists a random variable
X such that
E X < \infty and
, X_i, \le_\mathrm X for all
i \in I, where
\le_\mathrm denotes the increasing convex stochastic order defined by
A \le_\mathrm B if
E \phi(A) \le E \phi(B) for all nondecreasing convex real functions
\phi.
Relation to convergence of random variables
A sequence
\ converges to
X in the
L_1 norm if and only if it
converges in measure to
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
In measure theory, Lebesgue's dominated convergence theorem gives a mild sufficient condition under which limits and integrals of a sequence of functions can be interchanged. More technically it says that if a sequence of functions is bounded i ...
, see
Vitali convergence theorem
In real analysis and measure theory, the Vitali convergence theorem, named after the Italian mathematician Giuseppe Vitali, is a generalization of the better-known dominated convergence theorem of Henri Lebesgue. It is a characterization of th ...
.
Citations
References
*
* Diestel, J. and Uhl, J. (1977). ''Vector measures'', Mathematical Surveys 15, American Mathematical Society, Providence, RI {{isbn, 978-0-8218-1515-1
Martingale theory>X_m, : (0, 1/n)=1 for all
m \ge n .
* If
X is a UI random variable, by splitting
\operatorname E(, X, ) = \operatorname E(, X, I_)+\operatorname E(, X, I_) and bounding each of the two, it can be seen that a uniformly integrable random variable is always bounded in
L^1.
* If any sequence of random variables
X_n is dominated by an integrable, non-negative
Y: that is, for all ''ω'' and ''n'',
, X_n(\omega), \le Y(\omega),\ Y(\omega)\ge 0,\ \operatorname E(Y) < \infty, then the class
\mathcal of random variables
\ is uniformly integrable.
* A class of random variables bounded in
L^p (
p > 1) is uniformly integrable.
Relevant theorems
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(\mu).
*
Nelson Dunford, Dunford–
Pettis theoremA class of random variables
X_n \subset L^1(\mu) is uniformly integrable
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
it is
relatively compact
In mathematics, a relatively compact subspace (or relatively compact subset, or precompact subset) of a topological space is a subset whose closure is compact.
Properties
Every subset of a compact topological space is relatively compact (sinc ...
for the
weak topology
In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a ...
\sigma(L^1,L^\infty).
*
de la Vallée-Poussin theoremThe family
\_ \subset L^1(\mu) is uniformly integrable if and only if there exists a non-negative increasing convex function
G(t) such that
\lim_ \frac t = \infty \text \sup_\alpha \operatorname E(G(, X_, )) < \infty.
Uniform integrability and stochastic ordering
A family of random variables
\_ is uniformly integrable if and only if
there exists a random variable
X such that
E X < \infty and
, X_i, \le_\mathrm X for all
i \in I, where
\le_\mathrm denotes the increasing convex stochastic order defined by
A \le_\mathrm B if
E \phi(A) \le E \phi(B) for all nondecreasing convex real functions
\phi.
Relation to convergence of random variables
A sequence
\ converges to
X in the
L_1 norm if and only if it
converges in measure to
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
In measure theory, Lebesgue's dominated convergence theorem gives a mild sufficient condition under which limits and integrals of a sequence of functions can be interchanged. More technically it says that if a sequence of functions is bounded i ...
, see
Vitali convergence theorem
In real analysis and measure theory, the Vitali convergence theorem, named after the Italian mathematician Giuseppe Vitali, is a generalization of the better-known dominated convergence theorem of Henri Lebesgue. It is a characterization of th ...
.
Citations
References
*
* Diestel, J. and Uhl, J. (1977). ''Vector measures'', Mathematical Surveys 15, American Mathematical Society, Providence, RI {{isbn, 978-0-8218-1515-1
Martingale theory>X_m, : (0, 1/n)=1 for all
m \ge n .
* If
X is a UI random variable, by splitting
\operatorname E(, X, ) = \operatorname E(, X, I_)+\operatorname E(, X, I_) and bounding each of the two, it can be seen that a uniformly integrable random variable is always bounded in
L^1.
* If any sequence of random variables
X_n is dominated by an integrable, non-negative
Y: that is, for all ''ω'' and ''n'',
, X_n(\omega), \le Y(\omega),\ Y(\omega)\ge 0,\ \operatorname E(Y) < \infty, then the class
\mathcal of random variables
\ is uniformly integrable.
* A class of random variables bounded in
L^p (
p > 1) is uniformly integrable.
Relevant theorems
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(\mu).
*
Nelson Dunford, Dunford–
Pettis theoremA class of random variables
X_n \subset L^1(\mu) is uniformly integrable
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
it is
relatively compact
In mathematics, a relatively compact subspace (or relatively compact subset, or precompact subset) of a topological space is a subset whose closure is compact.
Properties
Every subset of a compact topological space is relatively compact (sinc ...
for the
weak topology
In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a ...
\sigma(L^1,L^\infty).
*
de la Vallée-Poussin theoremThe family
\_ \subset L^1(\mu) is uniformly integrable if and only if there exists a non-negative increasing convex function
G(t) such that
\lim_ \frac t = \infty \text \sup_\alpha \operatorname E(G(, X_, )) < \infty.
Uniform integrability and stochastic ordering
A family of random variables
\_ is uniformly integrable if and only if
there exists a random variable
X such that
E X < \infty and
, X_i, \le_\mathrm X for all
i \in I, where
\le_\mathrm denotes the increasing convex stochastic order defined by
A \le_\mathrm B if
E \phi(A) \le E \phi(B) for all nondecreasing convex real functions
\phi.
Relation to convergence of random variables
A sequence
\ converges to
X in the
L_1 norm if and only if it
converges in measure to
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
In measure theory, Lebesgue's dominated convergence theorem gives a mild sufficient condition under which limits and integrals of a sequence of functions can be interchanged. More technically it says that if a sequence of functions is bounded i ...
, see
Vitali convergence theorem
In real analysis and measure theory, the Vitali convergence theorem, named after the Italian mathematician Giuseppe Vitali, is a generalization of the better-known dominated convergence theorem of Henri Lebesgue. It is a characterization of th ...
.
Citations
References
*
* Diestel, J. and Uhl, J. (1977). ''Vector measures'', Mathematical Surveys 15, American Mathematical Society, Providence, RI {{isbn, 978-0-8218-1515-1
Martingale theory>X_m, : (0, 1/n)=1 for all
m \ge n .
* If
X is a UI random variable, by splitting
\operatorname E(, X, ) = \operatorname E(, X, I_)+\operatorname E(, X, I_) and bounding each of the two, it can be seen that a uniformly integrable random variable is always bounded in
L^1.
* If any sequence of random variables
X_n is dominated by an integrable, non-negative
Y: that is, for all ''ω'' and ''n'',
, X_n(\omega), \le Y(\omega),\ Y(\omega)\ge 0,\ \operatorname E(Y) < \infty, then the class
\mathcal of random variables
\ is uniformly integrable.
* A class of random variables bounded in
L^p (
p > 1) is uniformly integrable.
Relevant theorems
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(\mu).
*
Nelson Dunford, Dunford–
Pettis theoremA class of random variables
X_n \subset L^1(\mu) is uniformly integrable
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
it is
relatively compact
In mathematics, a relatively compact subspace (or relatively compact subset, or precompact subset) of a topological space is a subset whose closure is compact.
Properties
Every subset of a compact topological space is relatively compact (sinc ...
for the
weak topology
In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a ...
\sigma(L^1,L^\infty).
*
de la Vallée-Poussin theoremThe family
\_ \subset L^1(\mu) is uniformly integrable if and only if there exists a non-negative increasing convex function
G(t) such that
\lim_ \frac t = \infty \text \sup_\alpha \operatorname E(G(, X_, )) < \infty.
Uniform integrability and stochastic ordering
A family of random variables
\_ is uniformly integrable if and only if
there exists a random variable
X such that
E X < \infty and
, X_i, \le_\mathrm X for all
i \in I, where
\le_\mathrm denotes the increasing convex stochastic order defined by
A \le_\mathrm B if
E \phi(A) \le E \phi(B) for all nondecreasing convex real functions
\phi.
Relation to convergence of random variables
A sequence
\ converges to
X in the
L_1 norm if and only if it
converges in measure to
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
In measure theory, Lebesgue's dominated convergence theorem gives a mild sufficient condition under which limits and integrals of a sequence of functions can be interchanged. More technically it says that if a sequence of functions is bounded i ...
, see
Vitali convergence theorem
In real analysis and measure theory, the Vitali convergence theorem, named after the Italian mathematician Giuseppe Vitali, is a generalization of the better-known dominated convergence theorem of Henri Lebesgue. It is a characterization of th ...
.
Citations
References
*
* Diestel, J. and Uhl, J. (1977). ''Vector measures'', Mathematical Surveys 15, American Mathematical Society, Providence, RI {{isbn, 978-0-8218-1515-1
Martingale theory>X, I_A\varepsilon
whenever
P(A)<\delta.
It is equivalent to uniform integrability if the measure is finite and has no atoms.
The term "uniform
absolute continuity
In calculus and real analysis, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between ...
" is not standard, but is used by some authors.
Related corollaries
The following results apply to the probabilistic definition.
* Definition 1 could be rewritten by taking the limits as
\lim_ \sup_ \operatorname E(, X, \,I_)=0.
* A non-UI sequence. Let
\Omega = ,1\subset \mathbb, and define
X_n(\omega) = \begin
n, & \omega\in (0,1/n), \\
0 , & \text \end Clearly
X_n\in L^1, and indeed
\operatorname E(, X_n, )=1\ , for all ''n''. However,
\operatorname E(, X_n, I_)= 1\ \text n \ge 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
L^1 norm of all
X_ns are 1 i.e., bounded. But the second clause does not hold as given any
\delta positive, there is an interval
(0, 1/n) with measure less than
\delta and
E for all m \ge n .
* If X is a UI random variable, by splitting \operatorname E(, X, ) = \operatorname E(, X, I_)+\operatorname E(, X, I_) and bounding each of the two, it can be seen that a uniformly integrable random variable is always bounded in L^1.
* If any sequence of random variables X_n is dominated by an integrable, non-negative Y: that is, for all ''ω'' and ''n'', , X_n(\omega), \le Y(\omega),\ Y(\omega)\ge 0,\ \operatorname E(Y) < \infty, then the class \mathcal of random variables \ is uniformly integrable.
* A class of random variables bounded in L^p (p > 1) is uniformly integrable.
Relevant theorems
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(\mu).
* Nelson Dunford, Dunford– Pettis theoremA class of random variables X_n \subset L^1(\mu) is uniformly integrable if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
it is relatively compact
In mathematics, a relatively compact subspace (or relatively compact subset, or precompact subset) of a topological space is a subset whose closure is compact.
Properties
Every subset of a compact topological space is relatively compact (sinc ...
for the weak topology
In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a ...
\sigma(L^1,L^\infty).
* de la Vallée-Poussin theoremThe family \_ \subset L^1(\mu) is uniformly integrable if and only if there exists a non-negative increasing convex function G(t) such that \lim_ \frac t = \infty \text \sup_\alpha \operatorname E(G(, X_, )) < \infty.
Uniform integrability and stochastic ordering
A family of random variables \_ is uniformly integrable if and only if
there exists a random variable
X such that E X < \infty and
, X_i, \le_\mathrm X for all i \in I, where
\le_\mathrm denotes the increasing convex stochastic order defined by
A \le_\mathrm B if E \phi(A) \le E \phi(B) for all nondecreasing convex real functions \phi.
Relation to convergence of random variables
A sequence \ converges to X in the L_1 norm if and only if it converges in measure to 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
In measure theory, Lebesgue's dominated convergence theorem gives a mild sufficient condition under which limits and integrals of a sequence of functions can be interchanged. More technically it says that if a sequence of functions is bounded i ...
, see Vitali convergence theorem
In real analysis and measure theory, the Vitali convergence theorem, named after the Italian mathematician Giuseppe Vitali, is a generalization of the better-known dominated convergence theorem of Henri Lebesgue. It is a characterization of th ...
.
Citations
References
*
* Diestel, J. and Uhl, J. (1977). ''Vector measures'', Mathematical Surveys 15, American Mathematical Society, Providence, RI {{isbn, 978-0-8218-1515-1
Martingale theory>X, I_A\varepsilon
whenever
P(A)<\delta.
It is equivalent to uniform integrability if the measure is finite and has no atoms.
The term "uniform
absolute continuity
In calculus and real analysis, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between ...
" is not standard, but is used by some authors.
Related corollaries
The following results apply to the probabilistic definition.
* Definition 1 could be rewritten by taking the limits as
\lim_ \sup_ \operatorname E(, X, \,I_)=0.
* A non-UI sequence. Let
\Omega = ,1\subset \mathbb, and define
X_n(\omega) = \begin
n, & \omega\in (0,1/n), \\
0 , & \text \end Clearly
X_n\in L^1, and indeed
\operatorname E(, X_n, )=1\ , for all ''n''. However,
\operatorname E(, X_n, I_)= 1\ \text n \ge 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
L^1 norm of all
X_ns are 1 i.e., bounded. But the second clause does not hold as given any
\delta positive, there is an interval
(0, 1/n) with measure less than
\delta and
E for all m \ge n .
* If X is a UI random variable, by splitting \operatorname E(, X, ) = \operatorname E(, X, I_)+\operatorname E(, X, I_) and bounding each of the two, it can be seen that a uniformly integrable random variable is always bounded in L^1.
* If any sequence of random variables X_n is dominated by an integrable, non-negative Y: that is, for all ''ω'' and ''n'', , X_n(\omega), \le Y(\omega),\ Y(\omega)\ge 0,\ \operatorname E(Y) < \infty, then the class \mathcal of random variables \ is uniformly integrable.
* A class of random variables bounded in L^p (p > 1) is uniformly integrable.
Relevant theorems
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(\mu).
* Nelson Dunford, Dunford– Pettis theoremA class of random variables X_n \subset L^1(\mu) is uniformly integrable if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
it is relatively compact
In mathematics, a relatively compact subspace (or relatively compact subset, or precompact subset) of a topological space is a subset whose closure is compact.
Properties
Every subset of a compact topological space is relatively compact (sinc ...
for the weak topology
In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a ...
\sigma(L^1,L^\infty).
* de la Vallée-Poussin theoremThe family \_ \subset L^1(\mu) is uniformly integrable if and only if there exists a non-negative increasing convex function G(t) such that \lim_ \frac t = \infty \text \sup_\alpha \operatorname E(G(, X_, )) < \infty.
Uniform integrability and stochastic ordering
A family of random variables \_ is uniformly integrable if and only if
there exists a random variable
X such that E X < \infty and
, X_i, \le_\mathrm X for all i \in I, where
\le_\mathrm denotes the increasing convex stochastic order defined by
A \le_\mathrm B if E \phi(A) \le E \phi(B) for all nondecreasing convex real functions \phi.
Relation to convergence of random variables
A sequence \ converges to X in the L_1 norm if and only if it converges in measure to 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
In measure theory, Lebesgue's dominated convergence theorem gives a mild sufficient condition under which limits and integrals of a sequence of functions can be interchanged. More technically it says that if a sequence of functions is bounded i ...
, see Vitali convergence theorem
In real analysis and measure theory, the Vitali convergence theorem, named after the Italian mathematician Giuseppe Vitali, is a generalization of the better-known dominated convergence theorem of Henri Lebesgue. It is a characterization of th ...
.
Citations
References
*
* Diestel, J. and Uhl, J. (1977). ''Vector measures'', Mathematical Surveys 15, American Mathematical Society, Providence, RI {{isbn, 978-0-8218-1515-1
Martingale theory>X_m, : (0, 1/n)=1 for all
m \ge n .
* If
X is a UI random variable, by splitting
\operatorname E(, X, ) = \operatorname E(, X, I_)+\operatorname E(, X, I_) and bounding each of the two, it can be seen that a uniformly integrable random variable is always bounded in
L^1.
* If any sequence of random variables
X_n is dominated by an integrable, non-negative
Y: that is, for all ''ω'' and ''n'',
, X_n(\omega), \le Y(\omega),\ Y(\omega)\ge 0,\ \operatorname E(Y) < \infty, then the class
\mathcal of random variables
\ is uniformly integrable.
* A class of random variables bounded in
L^p (
p > 1) is uniformly integrable.
Relevant theorems
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(\mu).
*
Nelson Dunford, Dunford–
Pettis theoremA class of random variables
X_n \subset L^1(\mu) is uniformly integrable
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
it is
relatively compact
In mathematics, a relatively compact subspace (or relatively compact subset, or precompact subset) of a topological space is a subset whose closure is compact.
Properties
Every subset of a compact topological space is relatively compact (sinc ...
for the
weak topology
In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a ...
\sigma(L^1,L^\infty).
*
de la Vallée-Poussin theoremThe family
\_ \subset L^1(\mu) is uniformly integrable if and only if there exists a non-negative increasing convex function
G(t) such that
\lim_ \frac t = \infty \text \sup_\alpha \operatorname E(G(, X_, )) < \infty.
Uniform integrability and stochastic ordering
A family of random variables
\_ is uniformly integrable if and only if
there exists a random variable
X such that
E X < \infty and
, X_i, \le_\mathrm X for all
i \in I, where
\le_\mathrm denotes the increasing convex stochastic order defined by
A \le_\mathrm B if
E \phi(A) \le E \phi(B) for all nondecreasing convex real functions
\phi.
Relation to convergence of random variables
A sequence
\ converges to
X in the
L_1 norm if and only if it
converges in measure to
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
In measure theory, Lebesgue's dominated convergence theorem gives a mild sufficient condition under which limits and integrals of a sequence of functions can be interchanged. More technically it says that if a sequence of functions is bounded i ...
, see
Vitali convergence theorem
In real analysis and measure theory, the Vitali convergence theorem, named after the Italian mathematician Giuseppe Vitali, is a generalization of the better-known dominated convergence theorem of Henri Lebesgue. It is a characterization of th ...
.
Citations
References
*
* Diestel, J. and Uhl, J. (1977). ''Vector measures'', Mathematical Surveys 15, American Mathematical Society, Providence, RI {{isbn, 978-0-8218-1515-1
Martingale theory>X, I_A\varepsilon
whenever
P(A)<\delta.
It is equivalent to uniform integrability if the measure is finite and has no atoms.
The term "uniform
absolute continuity
In calculus and real analysis, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between ...
" is not standard, but is used by some authors.
Related corollaries
The following results apply to the probabilistic definition.
* Definition 1 could be rewritten by taking the limits as
\lim_ \sup_ \operatorname E(, X, \,I_)=0.
* A non-UI sequence. Let
\Omega = ,1\subset \mathbb, and define
X_n(\omega) = \begin
n, & \omega\in (0,1/n), \\
0 , & \text \end Clearly
X_n\in L^1, and indeed
\operatorname E(, X_n, )=1\ , for all ''n''. However,
\operatorname E(, X_n, I_)= 1\ \text n \ge 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
L^1 norm of all
X_ns are 1 i.e., bounded. But the second clause does not hold as given any
\delta positive, there is an interval
(0, 1/n) with measure less than
\delta and
E for all m \ge n .
* If X is a UI random variable, by splitting \operatorname E(, X, ) = \operatorname E(, X, I_)+\operatorname E(, X, I_) and bounding each of the two, it can be seen that a uniformly integrable random variable is always bounded in L^1.
* If any sequence of random variables X_n is dominated by an integrable, non-negative Y: that is, for all ''ω'' and ''n'', , X_n(\omega), \le Y(\omega),\ Y(\omega)\ge 0,\ \operatorname E(Y) < \infty, then the class \mathcal of random variables \ is uniformly integrable.
* A class of random variables bounded in L^p (p > 1) is uniformly integrable.
Relevant theorems
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(\mu).
* Nelson Dunford, Dunford– Pettis theoremA class of random variables X_n \subset L^1(\mu) is uniformly integrable if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
it is relatively compact
In mathematics, a relatively compact subspace (or relatively compact subset, or precompact subset) of a topological space is a subset whose closure is compact.
Properties
Every subset of a compact topological space is relatively compact (sinc ...
for the weak topology
In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a ...
\sigma(L^1,L^\infty).
* de la Vallée-Poussin theoremThe family \_ \subset L^1(\mu) is uniformly integrable if and only if there exists a non-negative increasing convex function G(t) such that \lim_ \frac t = \infty \text \sup_\alpha \operatorname E(G(, X_, )) < \infty.
Uniform integrability and stochastic ordering
A family of random variables \_ is uniformly integrable if and only if
there exists a random variable
X such that E X < \infty and
, X_i, \le_\mathrm X for all i \in I, where
\le_\mathrm denotes the increasing convex stochastic order defined by
A \le_\mathrm B if E \phi(A) \le E \phi(B) for all nondecreasing convex real functions \phi.
Relation to convergence of random variables
A sequence \ converges to X in the L_1 norm if and only if it converges in measure to 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
In measure theory, Lebesgue's dominated convergence theorem gives a mild sufficient condition under which limits and integrals of a sequence of functions can be interchanged. More technically it says that if a sequence of functions is bounded i ...
, see Vitali convergence theorem
In real analysis and measure theory, the Vitali convergence theorem, named after the Italian mathematician Giuseppe Vitali, is a generalization of the better-known dominated convergence theorem of Henri Lebesgue. It is a characterization of th ...
.
Citations
References
*
* Diestel, J. and Uhl, J. (1977). ''Vector measures'', Mathematical Surveys 15, American Mathematical Society, Providence, RI {{isbn, 978-0-8218-1515-1
Martingale theory>X, I_A\varepsilon
whenever
P(A)<\delta.
It is equivalent to uniform integrability if the measure is finite and has no atoms.
The term "uniform
absolute continuity
In calculus and real analysis, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between ...
" is not standard, but is used by some authors.
Related corollaries
The following results apply to the probabilistic definition.
* Definition 1 could be rewritten by taking the limits as
\lim_ \sup_ \operatorname E(, X, \,I_)=0.
* A non-UI sequence. Let
\Omega = ,1\subset \mathbb, and define
X_n(\omega) = \begin
n, & \omega\in (0,1/n), \\
0 , & \text \end Clearly
X_n\in L^1, and indeed
\operatorname E(, X_n, )=1\ , for all ''n''. However,
\operatorname E(, X_n, I_)= 1\ \text n \ge 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
L^1 norm of all
X_ns are 1 i.e., bounded. But the second clause does not hold as given any
\delta positive, there is an interval
(0, 1/n) with measure less than
\delta and
E X_m, : (0, 1/n)=1 for all
m \ge n .
* If
X is a UI random variable, by splitting
\operatorname E(, X, ) = \operatorname E(, X, I_)+\operatorname E(, X, I_) and bounding each of the two, it can be seen that a uniformly integrable random variable is always bounded in
L^1.
* If any sequence of random variables
X_n is dominated by an integrable, non-negative
Y: that is, for all ''ω'' and ''n'',
, X_n(\omega), \le Y(\omega),\ Y(\omega)\ge 0,\ \operatorname E(Y) < \infty, then the class
\mathcal of random variables
\ is uniformly integrable.
* A class of random variables bounded in
L^p (
p > 1) is uniformly integrable.
Relevant theorems
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(\mu).
*
Nelson Dunford, Dunford–
Pettis theoremA class of random variables
X_n \subset L^1(\mu) is uniformly integrable
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
it is
relatively compact
In mathematics, a relatively compact subspace (or relatively compact subset, or precompact subset) of a topological space is a subset whose closure is compact.
Properties
Every subset of a compact topological space is relatively compact (sinc ...
for the
weak topology
In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a ...
\sigma(L^1,L^\infty).
*
de la Vallée-Poussin theoremThe family
\_ \subset L^1(\mu) is uniformly integrable if and only if there exists a non-negative increasing convex function
G(t) such that
\lim_ \frac t = \infty \text \sup_\alpha \operatorname E(G(, X_, )) < \infty.
Uniform integrability and stochastic ordering
A family of random variables
\_ is uniformly integrable if and only if
there exists a random variable
X such that
E X < \infty and
, X_i, \le_\mathrm X for all
i \in I, where
\le_\mathrm denotes the increasing convex stochastic order defined by
A \le_\mathrm B if
E \phi(A) \le E \phi(B) for all nondecreasing convex real functions
\phi.
Relation to convergence of random variables
A sequence
\ converges to
X in the
L_1 norm if and only if it
converges in measure to
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
In measure theory, Lebesgue's dominated convergence theorem gives a mild sufficient condition under which limits and integrals of a sequence of functions can be interchanged. More technically it says that if a sequence of functions is bounded i ...
, see
Vitali convergence theorem
In real analysis and measure theory, the Vitali convergence theorem, named after the Italian mathematician Giuseppe Vitali, is a generalization of the better-known dominated convergence theorem of Henri Lebesgue. It is a characterization of th ...
.
Citations
References
*
* Diestel, J. and Uhl, J. (1977). ''Vector measures'', Mathematical Surveys 15, American Mathematical Society, Providence, RI {{isbn, 978-0-8218-1515-1
Martingale theory>X_m, : (0, 1/n)=1 for all
m \ge n .
* If
X is a UI random variable, by splitting
\operatorname E(, X, ) = \operatorname E(, X, I_)+\operatorname E(, X, I_) and bounding each of the two, it can be seen that a uniformly integrable random variable is always bounded in
L^1.
* If any sequence of random variables
X_n is dominated by an integrable, non-negative
Y: that is, for all ''ω'' and ''n'',
, X_n(\omega), \le Y(\omega),\ Y(\omega)\ge 0,\ \operatorname E(Y) < \infty, then the class
\mathcal of random variables
\ is uniformly integrable.
* A class of random variables bounded in
L^p (
p > 1) is uniformly integrable.
Relevant theorems
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(\mu).
*
Nelson Dunford, Dunford–
Pettis theoremA class of random variables
X_n \subset L^1(\mu) is uniformly integrable
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
it is
relatively compact
In mathematics, a relatively compact subspace (or relatively compact subset, or precompact subset) of a topological space is a subset whose closure is compact.
Properties
Every subset of a compact topological space is relatively compact (sinc ...
for the
weak topology
In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a ...
\sigma(L^1,L^\infty).
*
de la Vallée-Poussin theoremThe family
\_ \subset L^1(\mu) is uniformly integrable if and only if there exists a non-negative increasing convex function
G(t) such that
\lim_ \frac t = \infty \text \sup_\alpha \operatorname E(G(, X_, )) < \infty.
Uniform integrability and stochastic ordering
A family of random variables
\_ is uniformly integrable if and only if
there exists a random variable
X such that
E X < \infty and
, X_i, \le_\mathrm X for all
i \in I, where
\le_\mathrm denotes the increasing convex stochastic order defined by
A \le_\mathrm B if
E \phi(A) \le E \phi(B) for all nondecreasing convex real functions
\phi.
Relation to convergence of random variables
A sequence
\ converges to
X in the
L_1 norm if and only if it
converges in measure to
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
In measure theory, Lebesgue's dominated convergence theorem gives a mild sufficient condition under which limits and integrals of a sequence of functions can be interchanged. More technically it says that if a sequence of functions is bounded i ...
, see
Vitali convergence theorem
In real analysis and measure theory, the Vitali convergence theorem, named after the Italian mathematician Giuseppe Vitali, is a generalization of the better-known dominated convergence theorem of Henri Lebesgue. It is a characterization of th ...
.
Citations
References
*
* Diestel, J. and Uhl, J. (1977). ''Vector measures'', Mathematical Surveys 15, American Mathematical Society, Providence, RI {{isbn, 978-0-8218-1515-1
Martingale theory>X_m, : (0, 1/n)=1 for all
m \ge n .
* If
X is a UI random variable, by splitting
\operatorname E(, X, ) = \operatorname E(, X, I_)+\operatorname E(, X, I_) and bounding each of the two, it can be seen that a uniformly integrable random variable is always bounded in
L^1.
* If any sequence of random variables
X_n is dominated by an integrable, non-negative
Y: that is, for all ''ω'' and ''n'',
, X_n(\omega), \le Y(\omega),\ Y(\omega)\ge 0,\ \operatorname E(Y) < \infty, then the class
\mathcal of random variables
\ is uniformly integrable.
* A class of random variables bounded in
L^p (
p > 1) is uniformly integrable.
Relevant theorems
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(\mu).
*
Nelson Dunford, Dunford–
Pettis theoremA class of random variables
X_n \subset L^1(\mu) is uniformly integrable
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
it is
relatively compact
In mathematics, a relatively compact subspace (or relatively compact subset, or precompact subset) of a topological space is a subset whose closure is compact.
Properties
Every subset of a compact topological space is relatively compact (sinc ...
for the
weak topology
In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a ...
\sigma(L^1,L^\infty).
*
de la Vallée-Poussin theoremThe family
\_ \subset L^1(\mu) is uniformly integrable if and only if there exists a non-negative increasing convex function
G(t) such that
\lim_ \frac t = \infty \text \sup_\alpha \operatorname E(G(, X_, )) < \infty.
Uniform integrability and stochastic ordering
A family of random variables
\_ is uniformly integrable if and only if
there exists a random variable
X such that
E X < \infty and
, X_i, \le_\mathrm X for all
i \in I, where
\le_\mathrm denotes the increasing convex stochastic order defined by
A \le_\mathrm B if
E \phi(A) \le E \phi(B) for all nondecreasing convex real functions
\phi.
Relation to convergence of random variables
A sequence
\ converges to
X in the
L_1 norm if and only if it
converges in measure to
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
In measure theory, Lebesgue's dominated convergence theorem gives a mild sufficient condition under which limits and integrals of a sequence of functions can be interchanged. More technically it says that if a sequence of functions is bounded i ...
, see
Vitali convergence theorem
In real analysis and measure theory, the Vitali convergence theorem, named after the Italian mathematician Giuseppe Vitali, is a generalization of the better-known dominated convergence theorem of Henri Lebesgue. It is a characterization of th ...
.
Citations
References
*
* Diestel, J. and Uhl, J. (1977). ''Vector measures'', Mathematical Surveys 15, American Mathematical Society, Providence, RI {{isbn, 978-0-8218-1515-1
Martingale theory>X_m, : (0, 1/n)=1 for all
m \ge n .
* If
X is a UI random variable, by splitting
\operatorname E(, X, ) = \operatorname E(, X, I_)+\operatorname E(, X, I_) and bounding each of the two, it can be seen that a uniformly integrable random variable is always bounded in
L^1.
* If any sequence of random variables
X_n is dominated by an integrable, non-negative
Y: that is, for all ''ω'' and ''n'',
, X_n(\omega), \le Y(\omega),\ Y(\omega)\ge 0,\ \operatorname E(Y) < \infty, then the class
\mathcal of random variables
\ is uniformly integrable.
* A class of random variables bounded in
L^p (
p > 1) is uniformly integrable.
Relevant theorems
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(\mu).
*
Nelson Dunford, Dunford–
Pettis theoremA class of random variables
X_n \subset L^1(\mu) is uniformly integrable
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
it is
relatively compact
In mathematics, a relatively compact subspace (or relatively compact subset, or precompact subset) of a topological space is a subset whose closure is compact.
Properties
Every subset of a compact topological space is relatively compact (sinc ...
for the
weak topology
In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a ...
\sigma(L^1,L^\infty).
*
de la Vallée-Poussin theoremThe family
\_ \subset L^1(\mu) is uniformly integrable if and only if there exists a non-negative increasing convex function
G(t) such that
\lim_ \frac t = \infty \text \sup_\alpha \operatorname E(G(, X_, )) < \infty.
Uniform integrability and stochastic ordering
A family of random variables
\_ is uniformly integrable if and only if
there exists a random variable
X such that
E X < \infty and
, X_i, \le_\mathrm X for all
i \in I, where
\le_\mathrm denotes the increasing convex stochastic order defined by
A \le_\mathrm B if
E \phi(A) \le E \phi(B) for all nondecreasing convex real functions
\phi.
Relation to convergence of random variables
A sequence
\ converges to
X in the
L_1 norm if and only if it
converges in measure to
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
In measure theory, Lebesgue's dominated convergence theorem gives a mild sufficient condition under which limits and integrals of a sequence of functions can be interchanged. More technically it says that if a sequence of functions is bounded i ...
, see
Vitali convergence theorem
In real analysis and measure theory, the Vitali convergence theorem, named after the Italian mathematician Giuseppe Vitali, is a generalization of the better-known dominated convergence theorem of Henri Lebesgue. It is a characterization of th ...
.
Citations
References
*
* Diestel, J. and Uhl, J. (1977). ''Vector measures'', Mathematical Surveys 15, American Mathematical Society, Providence, RI {{isbn, 978-0-8218-1515-1
Martingale theory