# 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 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

Let be a positive measure space. If

- is uniformly integrable
- a.e. as and
- a.e.

then the following hold:

- .
^{[1]}

## Outline of Proof

- 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

Let be a positive measure space. If

- ,
- and
- exists for every

then is uniformly integrable.^{[1]}

## Citations

## References

- 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

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