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
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).
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
- exists for every
then is uniformly integrable.
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