There are two kinds of vectorvalued measures which are involved in the theory of weakly stationary processes: orthogonal (Hilbert space valued) and multiplicative (projection-valued) measures. For both classes we show that weak convergence is equivalent with the convergence of integrals over bounded continuous functions. Moreover we prove continuity theorems for the Fourier transformation as well as for the Laplace transformation of such measures.
"Weak Convergence of Certain Vectorvalued Measures." Ann. Probab. 2 (1) 136 - 142, February, 1974. https://doi.org/10.1214/aop/1176996758