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 $\mathscr{S}'(\mathbb{R}^d)$ of temperate real-valued Schwartz distributions on $\mathbb{R}^d$. It will be equipped with the strong topology and the corresponding Borel $\sigma$-algebra.
Let us call a function $\Phi:\mathscr{S}(\mathbb{R}^d)\rightarrow \mathbb{C}$ a characteristic function iff it satisfies the following three conditions:
- $\Phi(0)=1$,
- $\Phi$ is continuous for the usual Fréchet topology of the Schwartz space $\mathscr{S}(\mathbb{R}^d)$ of rapidly decaying real-valued test functions,
- for all $n\ge 1$ and all $f_1,\ldots,f_n$ in $\mathscr{S}(\mathbb{R}^d)$, 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 $\mathscr{S}'(\mathbb{R}^d)$, we define the characteristic function of $\mu$ as the function $\Phi_{\mu}:\mathscr{S}(\mathbb{R}^d)\rightarrow \mathbb{C}$ defined by $$ \Phi_{\mu}(f)=\int_{\mathscr{S}'(\mathbb{R}^d)}e^{i\varphi(f)}\ d\mu(\varphi)\ . $$
Theorem: (Bochner-Minlos) The map $\mu\mapsto\Phi_{\mu}$ is bijection from the set of Borel probability measures on $\mathscr{S}'(\mathbb{R}^d)$ to the set of characteristic functions.
The weak convergence of probability measures on $\mathscr{S}'(\mathbb{R}^d)$ is the same as for any other topological space.
Definition: Let $\mu$ be a Borel probability measure on $\mathscr{S}'(\mathbb{R}^d)$ and $(\mu_n)_{n\ge 1}$ be a sequence of Borel probability measures on $\mathscr{S}'(\mathbb{R}^d)$. We say that $\mu_n$ converges weakly to $\mu$ iff for all bounded continuous functions $F:\mathscr{S}'(\mathbb{R}^d)\rightarrow\mathbb{R}$, $$ \lim_{n\rightarrow\infty}\int_{\mathscr{S}'(\mathbb{R}^d)}F(\varphi)\ d\mu_n(\varphi) =\int_{\mathscr{S}'(\mathbb{R}^d)}F(\varphi)\ d\mu(\varphi)\ . $$
Now we 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, $$ \forall f\in\mathscr{S}(\mathbb{R}^d),\ \lim_{n\rightarrow\infty}\Phi_{\mu_n}(f)=\Phi_{\mu}(f)\ . $$
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 $\mathscr{S}'(\mathbb{R}^d)$. The existence of a Borel probability measure $\mu$ such that $\mu_n$ converges weakly to $\mu$, is equivalent to requiring
- For all $f\in\mathscr{S}(\mathbb{R}^d)$, 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 for now, but there are also other results like $\mathscr{S}'(\mathbb{R}^d)$ being a Radon space as well as Prokhorov's Theorem.