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 strong condition that depends on uniform integrability. It is useful when a dominating function cannot be found for the sequence of functions in question; when such a dominating function can be found, Lebesgue's theorem follows as a special case of Vitali's.

Let ${\displaystyle (X,{\mathcal {F}},\mu )}$ be a positive measure space. If

then the following hold:

Let ${\displaystyle (X,{\mathcal {F}},\mu )}$ be a positive measure space. If

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