The weak convergence is a very powerful tool
in Probability Theory, partly due to its comparative simplicity and partly due to
its natural behavior in some typical problems.The concept of weak convergence is so well established in Probability Theory
that hardly any textbook even mention its topological heritage. It, indeed, is not
too important in many applications, but a complete grasp of the definition of the weak convergence is not possible without understanding its rationale.
The first
part of this paper (Sections 2 and 3) is an introduction to weak convergence of
probability measures from the topological point of view. Since the set of probability
measures is not closed under weak convergence (as we shall see, the limit of a
net of probability measures need not be a probability measure), for a full understanding
of the complete concept, one has to investigate a wider structure, which
turns out to be the set of all finitely additive Radon measures. In this context we
present results concerning the Baire field and sigma field, which are usually omitted
when discussing probability measures. In Section 4 we consider weak convergence
of probability measures and present classical results regarding metrics of weak convergence.
In Section 5 we show that the set of probability measures is not closed
and effectively show the existence of a finitely, but not countably additive measure
in the closure of the set of probability measures. Section 6 deals with the famous
Prohorov's theorem on metric spaces. In Section 7 we consider weak convergence of
probability measures on Hilbert spaces. Here we observe a separable Hilbert space
equipped with weak and strong topology and in both cases we give necessary and
sufficient conditions for relative compactness of a set of probability measures.
