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

$$C_0^\infty(\Omega)$$ 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 ?

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.
• "one understands the space of finite signed measures on $S$ to be the dual of $C(S)$" Can you say more about this ? I fail to see it. Also, weak* convergence isn't just pointwise convergence ? Commented Dec 16, 2019 at 19:08
• "Loosely", I wrote. Read en.wikipedia.org/wiki/… for some more detail. The weak * topology on $C(S)*$ is indeed the topology of pointwise convergence, where the "points" are the elements of $C(S)$. Commented Dec 16, 2019 at 22:47