The authors introduce and study certain indices related to the Radon-Nikodým property in Banach spaces. Interesting quantitative versions of classic results in RNP are proved. Let \(E\) be a Banach space and \((\Omega,\Sigma,\mu)\) a complete probability space. Given a vector measure \(m:\Sigma \to E\), the index \(\mathcal R(m)\) of representability is defined. It turns out (Proposition 2.3) that \(\mathcal R(m)\) is the inf of all \(\delta>0\) for which there exists \(g\in L^1(\mu,E)\) such that \(\|m(A)-\int_A g \|\leq \delta\mu(A)\) for all \(A \in\Sigma\). The index of dentability \(\mathrm{Dent}(C)\) of a subset \(C\) of \(E\) is defined as the inf of \(\varepsilon>0\) for which \(C\) has non-empty slices contained in balls of radius \(< \varepsilon\). The measure of weak non-compactness is \(\gamma(C)= \sup\{\lim_n\lim_m x_m^*(x_n)- \lim_m\lim_n x_m^*(x_n)\}\), where the sup is taken over all \(x_m^*\) in the unit ball of the dual of \(E\) and \(x_n \in C\) for which the limits exist. It is shown that \(\mathrm{Dent}(C) \leq\gamma(C)\) (Prop. 6.1) and \(\sup\{\mathrm{Dent}(D),\,D\subseteq C\} \leq 2 \sup\{\mathrm{Dent}(D),\,D\subseteq C,\,D\text{ countable}\}\) (Prop. 4.6). The relation of the index of dentability and the Kuratowski measure of non-compactness is presented. In the case \(m:\Sigma\to E\) is defined by \(m(A)=T(\chi_A)\), where \(T:L^1\to E\) is a bounded operator, it is shown (Prop. 2.7) that \(\mathcal R(m)\) is equal to the distance of \(T\) (in the operator norm) from the space of Bochner representable operators from \(L^1\) to \(E\). If \(m\) takes values in a dual space \(X\), it is shown that \(\mathcal R(m)\) is equal to certain distances of the Gelfand derivative of \(m\) from the space of strongly measurable (or Bochner integrable) functions in \(X\). Examples of vector measures with values in \(c_0\) and \(L^1\) are given and an explicit calculation of \(\mathcal R(m)\) is made. In Section 5, a fragmentability index for subsets of a dual space is defined. The authors prove many nice results under the assumption that the fragmentability index of the set of the extreme points of a set \(C\) is positive. The authors claim that their approach gives extra insight to the classical results related to the RNP. The reviewer strongly agrees with this claim.