Confusion about Dual spaces, Schwartz distributions and measures (weak convergence)

Question

$C_0^\infty(\Omega)$ where $\Omega \subset \mathbb R^n$ is the space of real, smooth, compactly supported functions on $\Omega$. We equip it with the induced topology by the uniform convergence on $\Omega$ of all derivatives.

$D'(\Omega)$, the space of Schwartz distributions is its dual, that is all $u: C_0^\infty(\Omega) \rightarrow \mathbb C$ that are linear and continuous. Continuous is well defined as we equipped $C_0^\infty(\Omega)$ with a topology.

For $f_k, f \in C_0^\infty(\Omega)$, we say that $f_k \rightarrow f$ weakly if for all $u \in D'(\Omega)$, $u(f_k) \rightarrow u(f)$ in $\mathbb C $. All of this comes from an analysis course.

Now, in a probability course, we also talked about weak convergence. Let $(S,d)$ be a metric space space, $\mathcal B$ its Borel $\sigma$-algebra. Let $\mathcal Q$ be the space of all probability measures $Q : \mathcal B \rightarrow [0,1]$.

The confusing definition is the following: $Q_n \rightarrow Q$ weakly iff $\int f dQ_n \rightarrow \int f dQ$ for all $f \in C_b(S)$, where $C_b(S)$ is the space of of all continuous bounded functions from $S$ to $\mathbb R$.

If the "weak" terminology coincide in analysis and in probability, that would mean that the dual of $\mathcal Q$ is the set of elements of the form $Q \rightarrow \int f dQ$ where $Q \in \mathcal Q$ and $f\in C_b(S)$. But we never defined a topology on $\mathcal Q$ so what does it mean to have a dual without a topology (ie what are continuous map from $\mathcal Q$ to $\mathbb C$) ? Is there a classical topology for probability measures ? Or maybe weak means different things in analysis and probability ?

Answer 1

You are right, the probability terminology is a little sloppy.

More properly, in probability theory "weak" convergence should be called "weak * convergence". Loosely, one understands the space of finite signed measures on $S$ to be the dual of $C(S)$, the continuous bounded functions on $S$. One equips the space of measures with the weak star topology, according to which for each $f\in C(S)$ the function $\mu\mapsto\int_Sfd\mu$ is continuous. When restricted to the probability measures on $S$, one recovers the usual convergence-in-distribution topology.