Vitali convergence theorem

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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 strong condition that depends on uniform integrability. It is useful when a dominating function cannot be found for the sequence of functions in question; when such a dominating function can be found, Lebesgue's theorem follows as a special case of Vitali's.

Statement of the theorem[edit]

Let be a positive measure space. If

  1. is uniformly integrable
  2. a.e. as and
  3. a.e.

then the following hold:

  1. .[1]

Outline of Proof[edit]

For proving statement 1, we use Fatou's lemma:
  • Using uniform integrability, we have where is a set such that
  • By Egorov's theorem, converges uniformly on the set . for a large and . Using triangle inequality,
  • Plugging the above bounds on the RHS of Fatou's lemma gives us statement 1.
For statement 2, use , where and .
  • The terms in the RHS are bounded respectively using Statement 1, uniform integrability of and Egorov's theorem for all .

Converse of the theorem[edit]

Let be a positive measure space. If

  1. ,
  2. and
  3. exists for every

then is uniformly integrable.[1]

Citations[edit]

  1. ^ a b Rudin, Walter (1986). Real and Complex Analysis. p. 133. ISBN 978-0-07-054234-1. 

References[edit]

  • Folland, Gerald B. (1999). Real analysis. Pure and Applied Mathematics (New York) (Second ed.). New York: John Wiley & Sons Inc. pp. xvi+386. ISBN 0-471-31716-0.  MR 1681462
  • Rosenthal, Jeffrey S. (2006). A first look at rigorous probability theory (Second ed.). Hackensack, NJ: World Scientific Publishing Co. Pte. Ltd. pp. xvi+219. ISBN 978-981-270-371-2.  MR 2279622

External links[edit]