While waiting for more context around the question, one can already mention the main definitions and tools for this topic.

I will assume the space on which these probability measures live is the space $\mathscr{S}'(\mathbb{R}^d,\mathbb{R})$ of temperate real-valued Schwartz distributions on $\mathbb{R}^d$. 
It is the $\mathbb{R}$-vector space given as the topological dual of the
space $\mathscr{S}(\mathbb{R}^d,\mathbb{R})$ of real-valued Schwartz functions.
The latter carries the usual Fréchet topology.
Recall that a subset $A\subset\mathscr{S}(\mathbb{R}^d,\mathbb{R})$ is bounded iff, for all continuous seminorms $\rho$ on $\mathscr{S}(\mathbb{R}^d,\mathbb{R})$
$$
\sup_{f\in A}\rho(f)\ <\ \infty\ .
$$
The space $\mathscr{S}'(\mathbb{R}^d,\mathbb{R})$ is equipped with its canonical/standard topology, namely the *strong topology* which is the locally convex topology defined by the seminorms
$$
\varphi\ \longmapsto\ \sup_{f\in A}|\varphi(f)|
$$
where $A$ ranges over all bounded subsets of $\mathscr{S}(\mathbb{R}^d,\mathbb{R})$. Finally, $\mathscr{S}'(\mathbb{R}^d,\mathbb{R})$ is turned into a measurable space thanks to the Borel $\sigma$-algebra for the strong topology.

Let us call a function $\Phi:\mathscr{S}(\mathbb{R}^d,\mathbb{R})\rightarrow \mathbb{C}$ *a* characteristic function iff it satisfies the following three conditions:

 1. $\Phi(0)=1$,
 2. $\Phi$ is continuous,
 3. for all $n\ge 1$ and all $f_1,\ldots,f_n$ in $\mathscr{S}(\mathbb{R}^d,\mathbb{R})$, the matrix $(\Phi(f_i-f_j))_{1\le i,j\le n}$ is Hermitian positive semidefinite.

There is no harm in changing 2) to just continuity at the origin.
Now for a Borel probability measure $\mu$ on $\mathscr{S}'(\mathbb{R}^d,\mathbb{R})$, we define *the* characteristic function *of* $\mu$ as the function $\Phi_{\mu}:\mathscr{S}(\mathbb{R}^d,\mathbb{R})\rightarrow \mathbb{C}$ defined by
$$
\Phi_{\mu}(f)=\int_{\mathscr{S}'(\mathbb{R}^d,\mathbb{R})}e^{i\varphi(f)}\ d\mu(\varphi)\ .
$$

**Theorem:** (Bochner-Minlos)
The map $\mu\mapsto\Phi_{\mu}$ is a bijection from the set of Borel probability measures on $\mathscr{S}'(\mathbb{R}^d,\mathbb{R})$ to the set of characteristic functions.

The weak convergence of probability measures on $\mathscr{S}'(\mathbb{R}^d,\mathbb{R})$ is defined in the same way as for any other topological space.

**Definition:** Let $\mu$ be a Borel probability measure on $\mathscr{S}'(\mathbb{R}^d,\mathbb{R})$ and $(\mu_n)_{n\ge 1}$ be a sequence of Borel probability measures on $\mathscr{S}'(\mathbb{R}^d,\mathbb{R})$. We say that $\mu_n$ *converges weakly to* $\mu$ iff
for all bounded continuous functions $F:\mathscr{S}'(\mathbb{R}^d,\mathbb{R})\rightarrow\mathbb{R}$,
$$
\lim_{n\rightarrow\infty}\int_{\mathscr{S}'(\mathbb{R}^d,\mathbb{R})}F(\varphi)\ d\mu_n(\varphi)
=\int_{\mathscr{S}'(\mathbb{R}^d,\mathbb{R})}F(\varphi)\ d\mu(\varphi)\ .
$$ 

We now have an analogue of *Glivenko's Theorem*.

**Theorem:**
The setting being the same as for the previous definition, the weak convergence of $\mu_n$ to $\mu$ is equivalent to the pointwise convergence of characteristic functions, namely,
$$
\forall f\in\mathscr{S}(\mathbb{R}^d,\mathbb{R}),\ \lim_{n\rightarrow\infty}\Phi_{\mu_n}(f)=\Phi_{\mu}(f)\ .
$$

Finally, we also have an analogue of the *L&eacute;vy Continuity Theorem*, when one does not *a priori* have a candidate for the weak limit.

**Theorem:**
Let $(\mu_n)_{n\ge 1}$ be a sequence of Borel probability measures on $\mathscr{S}'(\mathbb{R}^d,\mathbb{R})$. The existence of a Borel probability measure $\mu$ such that $\mu_n$ converges weakly to $\mu$, is equivalent to requiring that

 1. for all $f\in\mathscr{S}(\mathbb{R}^d,\mathbb{R})$, the limit $\lim_{n\rightarrow\infty}\Phi_{\mu_n}(f)$ exists, and
 2. the function $\Phi$ defined by $\Phi(f)=\lim_{n\rightarrow\infty}\Phi_{\mu_n}(f)$ is continuous at the origin.

I'll stop here this attempt at a *formulaire raisonn&eacute;*, for now, but there are also other useful results like $\mathscr{S}'(\mathbb{R}^d,\mathbb{R})$ being a Radon space as well as Prokhorov's Theorem.