Convergence of weighted partial sums when the limiting distribution is not necessarily Radon

Let Bw be a non-separable Banach space of real-valued functions endowed with a weighted sup-norm. We consider partial sum processes as random functions with values in Bw. We establish weak convergence statements for these processes via their weighted approximation in probability by an appropriate sequence of Gaussian random functions. The main result deals with convergence of distributions of certain functionals in the case when the Wiener measure is not necessarily a Radon measure on Bw.