Uniform integrability

(Redirected from Dunford–Pettis theorem)

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

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

Definition A: Let   be a positive measure space. 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[3] 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[4] 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

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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   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 variables is called uniformly integrable (UI) if for every   there exists   such that  , where   is the indicator function  .

Tightness and uniform integrability

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One consequence of uniformly integrability of a class   of random variables is that family of laws or distributions   is tight. That is, for each  , there exists   such that   for all  .[8]

This however, does not mean that the family of measures   is tight. (In any case, tightness would require a topology on   in order to be defined.)

Uniform absolute continuity

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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   is a probability space. A classed   of random variables is uniformly absolutely continuous with respect to   if for any  , there is   such that   whenever  .

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.[10][11]

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

  • Definition 1 could be rewritten by taking the limits as  
  • A non-UI sequence. Let  , and define   Clearly  , and indeed   for all n. However,   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   pointwise.
  • By using Definition 2 in the above example, it can be seen that the first clause is satisfied as   norm of all  s are 1 i.e., bounded. But the second clause does not hold as given any   positive, there is an interval   with measure less than   and   for all  .
  • If   is a UI random variable, by splitting   and bounding each of the two, it can be seen that a uniformly integrable random variable is always bounded in  .
  • If any sequence of random variables   is dominated by an integrable, non-negative  : that is, for all ω and n,   then the class   of random variables   is uniformly integrable.
  • A class of random variables bounded in   ( ) is uniformly integrable.

Relevant theorems

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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  .

  • DunfordPettis theorem[13][14]
    A class[clarification needed] of random variables   is uniformly integrable if and only if it is relatively compact for the weak topology  .[clarification needed][citation needed]
  • de la Vallée-Poussin theorem[15][16]
    The family   is uniformly integrable if and only if there exists a non-negative increasing convex function   such that  

Relation to convergence of random variables

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A sequence   converges to   in the   norm if and only if it converges in measure to   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

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  1. ^ Rudin, Walter (1987). Real and Complex Analysis (3 ed.). Singapore: McGraw–Hill Book Co. p. 133. ISBN 0-07-054234-1.
  2. ^ Royden, H.L. & Fitzpatrick, P.M. (2010). Real Analysis (4 ed.). Boston: Prentice Hall. p. 93. ISBN 978-0-13-143747-0.
  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. pp. 134–137. ISBN 978-1-84800-047-6.
  5. ^ Williams, David (1997). Probability with Martingales (Repr. ed.). Cambridge: Cambridge Univ. Press. pp. 126–132. ISBN 978-0-521-40605-5.
  6. ^ Gut, Allan (2005). Probability: A Graduate Course. Springer. pp. 214–218. ISBN 0-387-22833-0.
  7. ^ Bass, Richard F. (2011). Stochastic Processes. Cambridge: Cambridge University Press. pp. 356–357. ISBN 978-1-107-00800-7.
  8. ^ Gut 2005, p. 236.
  9. ^ Bass 2011, p. 356.
  10. ^ Benedetto, J. J. (1976). Real Variable and Integration. Stuttgart: B. G. Teubner. p. 89. ISBN 3-519-02209-5.
  11. ^ Burrill, C. W. (1972). Measure, Integration, and Probability. McGraw-Hill. p. 180. ISBN 0-07-009223-0.
  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). "The spaces Lp and spaces of measures". Measure Theory Volume I. Berlin Heidelberg: Springer-Verlag. p. 268. doi:10.1007/978-3-540-34514-5_4. ISBN 978-3-540-34513-8.

References

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