The space of real Borel measures $$\mathcal {M}(S)$$ M(S) on a metric space S under the flat norm (dual bounded Lipschitz norm), ordered by the cone $$\mathcal {M}_+(S)$$ 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 $$\mathcal {M}_+(S)$$ M+(S) are compact and semiflows on $$\mathcal {M}_+(S)$$ M+(S) are continuous under much weaker conditions. In turn, the flat norm offers new challenges because $$\mathcal {M}(S)$$ M(S) is rarely complete and $$\mathcal {M}_+(S)$$ M+(S) is only complete if S is complete. As illustrations serve the eigenvalue problem for bounded additive and order-preserving homogeneous maps on $$\mathcal {M}_+(S)$$ M+(S) and continuous semiflows. Both topics prepare for a dynamical systems theory on $$\mathcal {M}_+(S)$$ M+(S) .