Measures under the flat norm as ordered normed vector space
The space of all real Borel measures \(\mathcal{M}(S)\) on a metric space \(S\) is usually considered by the variational norm. In this paper the authors consider that space under the \textit{flat norm}; i.e., as a linear subspace of the dual of Lip\((S)\), the space of bounded Lipschitz continuous functions on \(S\) endowed by the usual Lipschitz norm. Such a norm is complete if and only if \(S\) is uniformly discrete (and in such case it is equivalent by the variational norm). Under the flat norm compactness and continuity of semiflows on \(\mathcal{M}(S)_+\) are established under weaker conditions. Completeness and convergence of sequence in \(\mathcal{M}(S)_+\) are also studied.
scientific article; zbMATH DE number 6861654
scientific article