, Volume 22, Issue 1, pp 105–138 | Cite as

Measures under the flat norm as ordered normed vector space

  • Piotr Gwiazda
  • Anna Marciniak-Czochra
  • Horst R. Thieme


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 \(\mathcal {M}_+(S)\).


Measures Ordered normed vector space Homogeneous maps Spectral radius Krein–Rutman theorem Dynamical systems 

Mathematics Subject Classification

Primary 28A33 28C15 46E27 47B65 47D06 47H07 47J10 58C40 Secondary 47N60 



The authors thank Azmy Ackleh, Stephan Luckhaus, and an anonymous referee for useful comments.


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Copyright information

© Springer International Publishing 2017

Authors and Affiliations

  • Piotr Gwiazda
    • 1
  • Anna Marciniak-Czochra
    • 2
  • Horst R. Thieme
    • 3
  1. 1.Institute of Applied Mathematics and MechanicsUniversity of WarsawWarsawPoland
  2. 2.Institute of Applied MathematicsUniversity of HeidelbergHeidelbergGermany
  3. 3.School of Mathematical and Statistical SciencesArizona State UniversityTempeUSA

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