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

Statement of the theorem

Let, with. Then, in if and only if we have
Remark: If is finite, then the second condition is trivially true. Also, and implies the uniform integrability of, and the uniform integrability of implies.

Outline of Proof

Converse of the theorem

Let be a positive measure space. If
  1. ,
  2. and
  3. exists for every
then is uniformly integrable.

Citations