  In this paper we formulate a theory of measure-valued linear transport
equations on networks. The building block of our approach is the
initial/boundary-value problem for the measure-valued linear transport equation
on a bounded interval, which is the prototype of an arc of the network. For
this problem we give an explicit representation formula of the solution, which
also considers the total mass flowing out of the interval. Then we construct
the global solution on the network by gluing all the measure-valued solutions
on the arcs by means of appropriate distribution rules at the vertexes. The
measure-valued approach makes our framework suitable to deal with multiscale
flows on networks, with the microscopic and macroscopic phases represented by
Lebesgue-singular and Lebesgue-absolutely continuous measures, respectively, in
time and space.
