Measures under the flat norm as ordered normed vector space

The space of real Borel measures (mathcal {M}(S)) on a metric space S under the flat norm (dual bounded Lipschitz norm), ordered by the cone (mathcal {M}_+(S)) of nonnegative measures, is considered from an ordered normed vector space perspective in order to apply the well-developed theory of this area. The flat norm is considered in place of the variation norm because subsets of (mathcal {M}_+(S)) are compact and semiflows on (mathcal {M}_+(S)) are continuous under much weaker conditions. In turn, the flat norm offers new challenges because (mathcal {M}(S)) is rarely complete and (mathcal {M}_+(S)) is only complete if S is complete. As illustrations serve the eigenvalue problem for bounded additive and order-preserving homogeneous maps on (mathcal {M}_+(S)) and continuous semiflows. Both topics prepare for a dynamical systems theory on (mathcal {M}_+(S)).