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.
be a measure space, i.e.
is a set function such that
All functions considered in the sequel will be functions
We adopt the following definitions according to Bogachev's terminology.
, 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.
, μ )
be a measure space with
, μ )
For a proof, see Bogachev's monograph "Measure Theory, Volume I".
, μ )
be a measure space and
, μ )
, μ )
, μ )
μ (
, 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.
be measure space.
and has uniformly absolutely continuous integrals.
In addition, there exists
is uniformly integrable.
For a proof, see Bogachev's monograph "Measure Theory, Volume I".