Finitely purely atomic measures: coincidence and rigidity properties (Q1851419)

The main results in this paper contain the equivalence of several facts concerning a complete finite measure \(\mu\), such as the following: (1) \(\mu\) is atomic and has a finite number of atoms, (2) \(L^1(\mu)\) (or \(L^\infty(\mu)\)) is reflexive, (3) \(L^\infty(\mu)\) has the Radon--Nikodým property, (4) weak-\(L^p(\mu) = L^p(\mu)\) if \(1\leq p<\infty\), and (5) every Banach--valued \(\mu\)--measurable function is Pettis integrable if and only if it is Dunford integrable (or Bochner integrable).