A variational McShane integral characterization of the Radon-Nikodým property (Q2843871)

It is proved that a Banach space \(X\) has the Radon-Nikodým property if and only if for every \(X\)-valued additive interval function \(\Phi \) that has absolutely continuous McShane variational measure there is a McShane integrable function \(f : [0, 1]\to X\) such that NEWLINE\[NEWLINE \Phi (I) = (\text{Mc}) \int \limits_I f NEWLINE\]NEWLINE for every interval \(I \subset [0, 1]\), and \(f\) is weakly equivalent to a measurable function \(g\).