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

Preliminary definitions

Let be a measure space, i.e. is a set function such that and is countably-additive. All functions considered in the sequel will be functions, where or. We adopt the following definitions according to Bogachev's terminology.

A set of functions is called uniformly integrable if, i.e .
A set of functions is said to have uniformly absolutely continuous integrals if, i.e.. This definition is sometimes used as a definition of uniform integrability. However, it differs from the definition of uniform integrability given above.

When, a set of functions is uniformly integrable if and only if it is bounded in and has uniformly absolutely continuous integrals. If, in addition, is atomless, then the uniform integrability is equivalent to the uniform absolute continuity of integrals.

Finite measure case

Let be a measure space with. Let and be an -measurable function. Then, the following are equivalent :

and converges to in ;
The sequence of functions converges in -measure to and is uniformly integrable ;

For a proof, see Bogachev's monograph "Measure Theory, Volume I".

Infinite measure case

Let be a measure space and. Let and. Then, converges to in if and only if the following holds :

The sequence of functions converges in -measure to ;
has uniformly absolutely continuous integrals;
For every, there exists such that and

When, the third condition becomes superfluous and the first two conditions give the usual form of Lebesgue-Vitali's convergence theorem originally stated for measure spaces with finite measure. In this case, one can show that conditions 1 and 2 imply that the sequence is uniformly integrable.

Converse of the theorem

Let be measure space. Let and assume that exists for every. Then, the sequence is bounded in and has uniformly absolutely continuous integrals. In addition, there exists such that for every.
When, this implies that is uniformly integrable.
For a proof, see Bogachev's monograph "Measure Theory, Volume I".