In many situations arising from physics and real-life, one needs to reconstruct a polynomial approximant starting from a set of measurements. In the classical theory of interpolation, such measurements are typically formalized in terms of point-wise evaluations. This perfectly fits the nature of the events when measuring scalar quantities, such as the temperature, but becomes extremely rough or even misleading when dealing with non-scalar quantities: the velocity, as well as more involved fields, such the electric and the magnetic ones, are prominent examples. In fact, besides the sole framework of interpolation theory, this has proven to be a significant achievement also in the devisement of mimetic and finite elements method (FEM)-based structure preserving methods. As a consequence, it is natural to expect that projectors onto the finite dimensional spaces will capture the essence of the field under investigation. In this talk we face the following challenge: what can we do if a collection of sensors detects the flux of a field, which is thus a non point-wise set of data? To answer the main question, we establish an interpolation theory (sometimes called histopolation) where results of unisolvence and stability are recovered. Further, we are able to characterize the norm of the relative projector with a quantity that resembles the usual Lebesgue constant, allowing for the identification of a bridge with some classical instances of interpolation and approximation theory.