Let X and Y be Hausdorff spaces and denote by M(X) and M(Y) the corresponding spaces of finite and non-negative Borel measures, endowed with the weak topology. A Borel map \(\phi\) : \(X\to Y\) induces the map \({\tilde \phi}\): M(X)\(\to M(Y)\). The author gives necessary and sufficient conditions for \({\tilde \phi}\) to be open.In case of \(\phi\) being a surjection between Suslin spaces, \({\tilde \phi}\) is open if and only if \(\phi\) is. Several examples complete the exposition.