[Math] the relation between weak convergence of measures and weak convergence from functional analysis

Question

To keep things simple, we assume $X$ to be a polish space (think of $X$ as $\mathbb{R}^n$ for example). Let's denote with $P(X)$ the space of all Borel probability measure on $X$. We say $\{\mu_n\}\subset P(X)$ converges weakly to $\mu\in P(X)$, denoted by $\mu_n\Rightarrow \mu$ if

$$\int fd\mu_n\to\int fd\mu,\forall f\in C_b(X)$$

where $C_b(X)$ is the space of all bounded continuous real valued functions. Let this definition be seen as the purely probabilistic one.

From functional analysis we also have a concept of weak convergence and weak topology. Let $E$ be a Banach space and denote by $E'$ the dual. Then by considering the family $\{\phi_f:f\in E'\}$, where $\phi_f:E\to\mathbb{R}$ is the linear functional $\phi_f(x):=\langle f,x\rangle$, the weak topology on $E$ is the coarsest topology which makes all $\phi_f$ continuous. One can prove $x_n\to x$ weakly (in weak topology) if and only if $\phi_f(x_n)\to\phi(x)$ for all $f\in E'$.

Since $X$ is Polish, we have that $P(X)$ is Polish too. Now I can define a continuous linear functional $\phi_f: P(X)\to\mathbb{R}$ for $f\in C_b(X)$ by

$$\phi_f(\mu):=\int fd\mu$$

Infact $\phi_f\in P(X)'$. Therefore we have $C_b(X)\subset P(X)'$.

My questions are

$\textbf{1. Question}$ Is the dual $P(X)'$ known? Is it isomorphic to a well known space? Is $C_b(X)$ a proper subspace?

$\textbf{2. Question}$ The notation of weak convergence in the probabilistic sense ($\mu_n\Rightarrow \mu$) is weaker than the weak convergence in the functional analytical sense. What I mean is: If $\mu_n\to\mu$ in the weak toplogy, i.e. $\langle f,\mu_n\rangle \to \langle f,\mu\rangle $ for all $f\in P(X)'$ this implies $\mu_n\Rightarrow \mu$ since $C_b(X)\subset P(X)'$. Is this the reason why one calls $\mu_n\Rightarrow \mu$ weak convergence, or is there any other reason?

Answer 1

"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.