What is the relation between weak convergence of measures and weak convergence from functional analysis

"Weak convergence of measures" is a misnomer. What it really means is that the space of measures is identified, via Riesz representation, with the dual of some space of continuous functions, and this gives us weak* topology on the space of measures. People just don't like saying that their measures converge weak-star-ly, or putting a lot of asterisks in their texts.

Folland writes in Real Analysis, page 223:

The weak* topology on $M(X)=C_0(X)^*$ ... is of considerable importance in applications; we shall call it the vague topology on $M(X)$. (The term "vague" is common in probability theory and has the advantage of forming an adverb more gracefully than "weak*".) The vague topology is sometimes called the weak topology, but this terminology conflicts with ours, since $C_0(X)$ is rarely reflexive.

Observe that the notation $P(X)'$ does not make sense as $P(X)$ is not a linear space. But you can topologize $P(X)$ in many ways:

1. as a subspace (in the topological sense) of $C_0(X)'$ with the norm topology, the distance between any two probabilities $\mu$ and $\nu$ being $\sup\big\{\big\vert\int f\,d\mu - \int f\,d\nu\big\vert:\;f\in C_0(X),\;\Vert f\Vert\leq1\big\}$
2. as a subspace of $C_0(X)'$ with its weak* topology, under which $\mu_n\rightarrow \mu$ if and only if $\int f\,d\mu_n\rightarrow\int f\,d\mu$ for all $f\in C_0(X)$.
3. as a subspace of $C_b(X)'$ with the norm topology, the distance between any two probabilities $\mu$ and $\nu$ being $\sup\big\{\big\vert\int f\,d\mu - \int f\,d\nu\big\vert:\;f\in C_b(X),\;\Vert f\Vert\leq1\big\}$
4. as a subspace of $C_b(X)'$ with its weak* topology, under which $\mu_n\rightarrow \mu$ if and only if $\int f\,d\mu_n\rightarrow\int f\,d\mu$ for all $f\in C_b(X)$. This is what probabilists usually call weak convergence.

If K denotes the Stone-Cech compactification βX of X, a description of M(K)' -viewed as the second dual of C(K) can be obtained by using the Arens product, which in the case of commutative C-* algebras implies that M(K)' is isometrically isomorphic to $C(\tilde K)$ -where $\tilde K$ is a compact (Stonean) space, called the hyperstonean envelope of $K$. There is a reecnt book devoted to such issues: H.G. Dales, F.K.Dashiell,Jr., A.T.M. Lau, D. Strauss : "Banach Spaces of Continuous Functions as Dual Space