  We investigate the well-posedness and approximation of mild solutions to a
class of linear transport equations on the unit interval $[0,1]$ endowed with a
linear discontinuous production term, formulated in the space
$\mathcal{M}([0,1])$ of finite Borel measures. Our working technique includes a
detailed boundary layer analysis in terms of a semigroup representation of
solutions in spaces of measures able to cope with the passage to the singular
limit where thickness of the layer vanishes. We obtain not only a suitable
concept of solutions to the chosen measure-valued evolution problem, but also
derive convergence rates for the approximation procedure and get insight in the
structure of flux boundary conditions for the limit problem.
