Let F Be Nonnegative Integrable and Uniformly Continuous

Mathematical concept

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.

Measure-theoretic definition [edit]

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 often use the following definition:^{[1] [2]}

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 slightly more general definition^[3] of uniform integrability that works well in general measures 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 ${\displaystyle L_{+}^{1}(\mu )=\{g\in L^{1}(\mu ):g\geq 0\}}$ .

For finite measure spaces the following result^[4] follows from Definition H:

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

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

Many textbooks in probability present Theorem 1 as the definition of uniform integrability in Probability spaces. When the space ${\displaystyle (X,{\mathfrak {M}},\mu )}$ is ${\displaystyle \sigma }$ -finite, Definition H yields the following equivalency:

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 surely. 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 exits ${\displaystyle \delta >0}$ such that

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

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

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

Probability definition [edit]

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

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

or alternatively

2. A class ${\displaystyle {\mathcal {C}}}$ of random variables is called uniformly integrable (UI) if 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 ${\displaystyle I_{|X|\geq K}={\begin{cases}1&{\text{if }}|X|\geq K,\\0&{\text{if }}|X|<K.\end{cases}}}$ .

Tightness and uniform integrability [edit]

One consequence of uniformly integrability of a class ${\displaystyle {\mathcal {C}}}$ of random variables is that family of laws or distributions ${\displaystyle \{P\circ |X|^{-1}(\cdot ):X\in {\mathcal {C}}\}}$ is tight. That is, for each ${\displaystyle \delta >0}$ , there exists ${\displaystyle a>0}$ such that

$${\displaystyle P(|X|>a)\leq \delta }$$

for all ${\displaystyle X\in {\mathcal {C}}}$ .^[8]

This however, does not mean that the family of measures ${\displaystyle {\mathcal {V}}_{\mathcal {C}}:={\Big \{}\mu _{X}:A\mapsto \int _{A}|X|\,dP,\,X\in {\mathcal {C}}{\Big \}}}$ is tight.

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^[9]

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

The term "uniform absolute continuity" is not standard, but is used by some other authors.^{[10] [11]}

[edit]

The following results apply to the probabilistic definition.^[12]

• 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,

  $${\displaystyle \operatorname {E} (|X_{n}|,|X_{n}|\geq K)=1\ {\text{ for all }}n\geq K,}$$

  

  and comparing with definition 1, it is seen that the sequence is not uniformly integrable.

Non-UI sequence of RVs. The area under the strip is always equal to 1, but ${\displaystyle X_{n}\to 0}$ pointwise.

• 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

  $${\displaystyle \operatorname {E} (|X|)=\operatorname {E} (|X|,|X|>K)+\operatorname {E} (|X|,|X|<K)}$$

  

  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.

Relevant theorems [edit]

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

Relation to convergence of random variables [edit]

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.

Citations [edit]

1. ^ Rudin, Walter (1987). Real and Complex Analysis (3 ed.). Singapore: McGraw–Hill Book Co. p. 133. ISBN0-07-054234-1.
2. ^ Royden, H.L. & Fitzpatrick, P.M. (2010). Real Analysis (4 ed.). Boston: Prentice Hall. p. 93. ISBN978-0-13-143747-0.
3. ^ Hunt, G. A. (1966). Martingales et Processus de Markov. Paris: Dunod. p. 254.
4. ^ Klenke, A. (2008). Probability Theory: A Comprehensive Course. Berlin: Springer Verlag. p. 134-137. ISBN978-1-84800-047-6.
5. ^ Williams, David (1997). Probability with Martingales (Repr. ed.). Cambridge: Cambridge Univ. Press. pp. 126–132. ISBN978-0-521-40605-5.
6. ^ Gut, Allan (2005). Probability: A Graduate Course. Springer. pp. 214–218. ISBN0-387-22833-0.
7. ^ Bass, Richard F. (2011). Stochastic Processes. Cambridge: Cambridge University Press. pp. 356–357. ISBN978-1-107-00800-7.
8. ^ Gut 2005, p. 236.
9. ^ Bass 2011, p. 356. sfn error: no target: CITEREFBass2011 (help)
10. ^ Benedetto, J. J. (1976). Real Variable and Integration. Stuttgart: B. G. Teubner. p. 89. ISBN3-519-02209-5.
11. ^ Burrill, C. W. (1972). Measure, Integration, and Probability. McGraw-Hill. p. 180. ISBN0-07-009223-0.
12. ^ Gut 2005, pp. 215–216.
13. ^ Dunford, Nelson (1938). "Uniformity in linear spaces". Transactions of the American Mathematical Society. 44 (2): 305–356. doi:10.1090/S0002-9947-1938-1501971-X. ISSN 0002-9947.
14. ^ Dunford, Nelson (1939). "A mean ergodic theorem". Duke Mathematical Journal. 5 (3): 635–646. doi:10.1215/S0012-7094-39-00552-1. ISSN 0012-7094.
15. ^ Meyer, P.A. (1966). Probability and Potentials, Blaisdell Publishing Co, N. Y. (p.19, Theorem T22).
16. ^ Poussin, C. De La Vallee (1915). "Sur L'Integrale de Lebesgue". Transactions of the American Mathematical Society. 16 (4): 435–501. doi:10.2307/1988879. hdl:10338.dmlcz/127627. JSTOR 1988879.
17. ^ Bogachev, Vladimir I. (2007). Measure Theory Volume I. Berlin Heidelberg: Springer-Verlag. p. 268. doi:10.1007/978-3-540-34514-5_4. ISBN978-3-540-34513-8.

References [edit]

• Shiryaev, A.N. (1995). Probability (2 ed.). New York: Springer-Verlag. pp. 187–188. ISBN978-0-387-94549-1.
• 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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Source: https://en.wikipedia.org/wiki/Uniform_integrability

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