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:
- $\Phi(0)=1$,
- $\Phi$ is continuous,
- 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 Lévy 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
- for all $f\in\mathscr{S}(\mathbb{R}^d,\mathbb{R})$, the limit $\lim_{n\rightarrow\infty}\Phi_{\mu_n}(f)$ exists, and
- 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 raisonné, 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.