The near Radon-Nikodym property in Lebesgue-Bochner function spaces (Q1129402)

Recall that a Banach space \(X\) has the near Radon-Nikodym property if every nearly representable linear operator \(A\) from \(L_1\) into \(X\) is representable. Here, \(A\) is called representable if \(Af=\int fg\) for some \(g\in L_\infty([0,1],X)\) and nearly representable if \(AD\) is representable for each Dunford-Pettis operator \(D\). The main result is that the space \(L_1(\lambda,X)\) with a finite measure space \(\lambda\) has the near Radon-Nikodym property if and only if \(X\) does. Moreover, if \(E\) is a Köthe function space which does not contain a copy of \(c_0\) and \(X\) has the near Radon-Nikodym property, then \(E(X)\) has the near Radon-Nikodym property.