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 characterization of the convergence in in terms of convergence in measure and a condition related to the Uniform integrability.

Statement of the theorem[edit]

Let , with . Then, in if and only if we have

  • (i) converge in measure to .
  • (ii) For every there exists a measurable set with such that for every disjoint from we have, for every
  • (iii) For every there exists such that, if and then, for every we have

Remark: If is finite, then the second condition is superfluous. Also, (i) and (iii) implies the uniform integrability of , and the uniform integrability of implies (iii). [1]

Outline of Proof[edit]

For proving statement 1, we use Fatou's lemma:
  • Using uniform integrability there exists such that we have for every set with
  • 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.[2]

Citations[edit]

  1. ^ SanMartin, Jaime (2016). Teoría de la medida. p. 280.
  2. ^ Rudin, Walter (1986). Real and Complex Analysis. p. 133. ISBN 978-0-07-054234-1.

References[edit]

  • Modern methods in the calculus of variations. 2007. ISBN 9780387357843.
  • 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. MR1681462
  • 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. MR2279622

External links[edit]