The set of nonnegative finite Borel measures on a metric space has been used as state space for dynamical systems modeling structured populations and other transport-related phenomena and stage-structured evolution.
It is the purpose of this talk to consider this set as cone of an ordered normed vector space of real measures and apply the rich theory of operators on such cones. While a familiar norm is given by the total variation, the flat norm is an attractive alternative because it makes transport semigroups continuous in time and offers convenient compactness conditions. But the flat norm also presents unusual challenges because it is hardly ever complete and is only complete on the cone of nonnegative measures if the underlying metric space is complete.
As illustrations serve continuous semiflows and the eigenvalue problem for order-preserving linear and homogeneous maps. Both topics are preparations for a dynamical systems theory on the nonnegative measures.