Weak sequential convergence in L1(μ,X) and an exact version of Fatouʼs lemma

The class of nonatomic finite measure spaces with the saturation property, as developed in Maharam (1942) and Hoover-Keisler (1984), is characterized by the Fatou (and Lebesgue) property of a well-dominated sequence of multifunctions taking values in a Banach space. With multifunctions reduced to functions, this Fatou characterization also extends to a variant of the closure property found in optimal control theory. The results are developed through a considered overview of the relevant literature on the exact and approximate Fatou lemma phrased in terms of Bochner integration.