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 the convergence in Lp in terms of convergence in measure and a condition related to uniform integrability.

Preliminary definitions

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Let   be a measure space, i.e.   is a set function such that   and   is countably-additive. All functions considered in the sequel will be functions  , where   or  . We adopt the following definitions according to Bogachev's terminology.[1]

  • A set of functions   is called uniformly integrable if  , i.e  .
  • A set of functions   is said to have uniformly absolutely continuous integrals if  , i.e.  . This definition is sometimes used as a definition of uniform integrability. However, it differs from the definition of uniform integrability given above.


When  , a set of functions   is uniformly integrable if and only if it is bounded in   and has uniformly absolutely continuous integrals. If, in addition,   is atomless, then the uniform integrability is equivalent to the uniform absolute continuity of integrals.

Finite measure case

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Let   be a measure space with  . Let   and   be an  -measurable function. Then, the following are equivalent :

  1.   and   converges to   in   ;
  2. The sequence of functions   converges in  -measure to   and   is uniformly integrable ;


For a proof, see Bogachev's monograph "Measure Theory, Volume I".[1]

Infinite measure case

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Let   be a measure space and  . Let   and  . Then,   converges to   in   if and only if the following holds :

  1. The sequence of functions   converges in  -measure to   ;
  2.   has uniformly absolutely continuous integrals;
  3. For every  , there exists   such that   and  

When  , the third condition becomes superfluous (one can simply take  ) and the first two conditions give the usual form of Lebesgue-Vitali's convergence theorem originally stated for measure spaces with finite measure. In this case, one can show that conditions 1 and 2 imply that the sequence   is uniformly integrable.

Converse of the theorem

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Let   be measure space. Let   and assume that   exists for every  . Then, the sequence   is bounded in   and has uniformly absolutely continuous integrals. In addition, there exists   such that   for every  .

When  , this implies that   is uniformly integrable.

For a proof, see Bogachev's monograph "Measure Theory, Volume I".[1]

Citations

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  1. ^ a b c Bogachev, Vladimir I. (2007). Measure Theory Volume I. New York: Springer. pp. 267–271. ISBN 978-3-540-34513-8.