TITLE: Topology of weak convergence
QUESTION [5 upvotes]: Edited: Thanks to etienne.
I start with a compact metric space $(X,d)$. Then I consider the collection of finite measure $\mathcal{M}$ on $X$ and I equip $\mathcal{M}$ with the topology of weak convergence. This means that $\mu_n$ converges to $\mu$ if and only if for all $f \in C(X, \mathbb{R})$ (continuous functions from $X$ to $\mathbb{R}$),
$$
\int f d\mu_n \to \int f d\mu.  
$$
Now my question is what does a general continuous function from $\mathcal{M}$ to $\mathbb{R}$ look like? By definition for $f \in C(X, \mathbb{R})$,
$$
F(\mu) := \int f d\mu
$$
is continuous but is also linear so this cannot be all continuous functions. Can we describe the entire set in some way?

REPLY [3 votes]: If you restrict yourself to $\mathcal M_1=\{ \mu\in\mathcal M;\; \Vert\mu\Vert\leq 1\}$, then any continuous function on $\mathcal M_1$ is a uniform limit of a sum of products of functions of the form $F(\mu)=\int f\, d\mu$. This follows from the Stone-Weierstrass theorem.