Fatou–Lebesgue theorem

In mathematics, the Fatou–Lebesgue theorem establishes a chain of inequalities relating the integrals (in the sense of Lebesgue) of the limit inferior and the limit superior of a sequence of functions to the limit inferior and the limit superior of integrals of these functions. The theorem is named after Pierre Fatou and Henri Léon Lebesgue.

If the sequence of functions converges pointwise, the inequalities turn into equalities and the theorem reduces to Lebesgue's dominated convergence theorem.