Topological properties of the addition map in spaces of Borel measures (Q1093758)

Given a topological space X, endow the space M(X) of non-negative, finite Borel measures on X with the weak (or narrow) topology. It is shown that without any assumption the map \(M(X)^ 2\ni (\nu _ 1,\nu _ 2)\to \nu _ 1+\nu _ 2\in M(X)\) is open and closed. As a main tool, the author proves a decomposition lemma for non-negative, finite measures on a finite set with respect to a relevant partial preordering.