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 Lp 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 \mu is countably-additive. All functions considered in the sequel will be functions, where or \mathbb{C}. We adopt the following definitions according to Bogachev's terminology. 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, \mu 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 f be an \mathcal{A}-measurable function. Then, the following are equivalent : For a proof, see Bogachev's monograph "Measure Theory, Volume I".

Infinite measure case

Let be a measure space and. Let and. Then, (f_n) converges to f in if and only if the following holds : When, the third condition becomes superfluous (one can simply take ) 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 (f_n) is bounded in and has uniformly absolutely continuous integrals. In addition, there exists such that for every. When, this implies that (f_n) is uniformly integrable. For a proof, see Bogachev's monograph "Measure Theory, Volume I".

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