Weak sequential convergence in \(L^1(\mu,X)\) and an exact version of Fatou's lemma (Q2019127)

The central result of the paper is Theorem 3.7 which roughly states that when \((\Omega, \mathcal{F},\mu)\) is a nonatomic finite measure space and \(X\) is an infinite-dimensional Banach space, then the following are equivalent: \newline (a) \((\Omega, \mathcal{F},\mu)\) has the saturation property of Maharam;\newline (b) a resonable generalization of Fatou's lemma holds. A corresponding result for multifunctions is proved in Theorem 4.9. The paper under review is also a survey of concepts involved in looking for generalization(s) of Fatou's lemma to the vector-valued case; a generalization that is important for applications. As the authors aptly describe their paper, ``[t]he results are developed through a considered overview of the relevant literature on the exact and approximate Fatou lemma phrased in terms of Bochner integration.''