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Topics in weak convergence of probability measures

The first part of the paper deals with general features of weak star convergence from a topological point of view. We present basic facts about weak and weak star topologies, dual spaces and representations of linear functionals as Radon measures. A special attention is payed to finitely additive measures and some results regarding the Baire sets. We prove that in a wide class of topological spaces (e.g. in non-compact metric spaces), the set of probability measures is not closed under weak star convergence, and we discuss measures in the closure. A classical theory of weak convergence of probability measures, including Prohorov's theorem is also presented. The second part of the paper is devoted to probability measures on separable Hilbert spaces. We discuss characteristic functions and its properties. The topic of weak compactness is studied assuming that a Hilbert space is equipped with the weak topology and with the norm topology separately. Here we also present a Prohorov's theorem on Hilbert spaces with the weak topology.