Weak convergence in spaces of measures and operators

J. K. Brooks and P. W. Lewis have established that if E and E* have RNP, then in M(Sigma, E), m(n) converges weakly to m if and only if m(n)(A) converges weakly to m(A) for each A is an element of Sigma. Assuming the existence of a special]kind of lifting, N. Randrianantoanina and E. Saab have shown an analogous result if E is a dual space. Here we show that for the space M(P(N) E) where E* is a Grothendieck space or E is a Mazur space, this kind of weak convergence is valid. Also some applications for subspaces of L(E, F) similar to the results of N. Kalton and W. Ruess are given.