Weak Convergence of Probability Measures on Metric Spaces of Nonlinear Operators

The conditions for weak convergence of a sequence of probability measures on metric spaces of nonlinear operators defined on some subsets of a real separable Banach space are established. The nonlinear operators of interest include either continuous operators or cadlag (continu à droite, limites à gauche) operators defined in this article. As the domains of the operators are certain compact sets, the limiting probability measures are the generalizations of the Wiener measure and the Poisson measure on the metric spaces of continuous and cadlag real functions defined on the unit interval, respectively. As the limiting probability measure is the generalized Wiener measure, the result is a generalization of Donsker’s theorem.