Approximation of integration maps of vector measures and limit representations of Banach function spaces

We study when the integration maps of vector measures can be computed as pointwise limits of their finite rank Radon-Nikod\'ym derivatives. We will show that this can sometimes be done, but there are also principal cases in which this cannot be done. The positive cases are obtained using the circle of ideas of the approximation property for Banach spaces. The negative ones are given by means of an adequate use of the Daugavet property. As an application, we analyse when the norm in a space of integrable functions $L^1(m)$ can be computed as a limit of the norms of the spaces of integrable functions with respect to the Radon-Nikod\'ym derivatives of $m$.