I am interested in a generalization of the following finite-dimensional results in infinite dimensional vector-space with nuclear structure, especially for the cases of the spaces of distributions $\mathcal{D}'(\mathrm{R}^N)$ and $\mathcal{S}'(\mathrm{R}^N)$.


**Theorem 1:** 
Let $X\_n$, $n\in\mathbb{N}$ and $X$ be random variables in $\mathbb{R}^N$ and $\Phi\_n$ and $\Phi$ their characteristic functions. Then,
$$\left( X\_n \overset{\mathcal{L}}{\rightarrow} X \right) \Leftrightarrow \left( \forall \omega, \Phi\_n(\omega) \rightarrow \Phi(\omega) \right).$$


**Theorem 2: (Lévy's continuity theorem)**
Again, the $X\_n$'s are random variable and the $\Phi\_n$'s are their characteristic functions. Assume that the limit $\lim \Phi\_n(\omega)$ exists pointwise and is denoted by $\Phi(\omega)$. We then have the equivalence:


* $(X\_n)$ converges in law to some random variable $X$.
* $\Phi$ is continuous at $0$.


Now I precise my question. Given a probability measure $\mu$ on $\mathcal{N}' = \mathcal{D}'(\mathrm{R}^N)$ or $\mathcal{S}'(\mathrm{R}^N)$, we can define its characteristic functional on $\mathcal{N}$ by 
$$\hat{\mu}(\varphi) = \int\_{\mathcal{N}'} \mathrm{e}^{\mathrm{j} \langle u , \varphi \rangle} \mathrm{d}\mu (u).$$
This generalizes the concept characteristic function of a random variable (Bochner's theorem being generalized by Minlos's theorem). Is the following result true? 


**Possible generalization of theorem 1:**
Let $\mu\_n$, $n\in\mathbb{N}$ and $\mu$ be probability measures on $\mathcal{N}'$ and $\hat{\mu}\_n$ and $\mu$ their characteristic functionals. Then,
$$\left( \mu\_n \overset{\mathrm{weakly}}{\rightarrow} \mu \right) \Leftrightarrow \left( \forall \varphi, \hat{\mu}\_n(\varphi) \rightarrow \mu(\varphi) \right).$$


Similarly, can we generalize the Lévy's continuity theorem?


Thanks for attention.



Generalization of Lévy's continuity theorem for nuclear spaces
There is a partial result due to Boulicaut (1973), which states 



> 
> **Theorem:** Let $E$ be a separable metrizable Hausdorff locally convex topological vector space. Then $E$ is nuclear if and only if for every sequence $\{\mu\_n\}$ of tight probability measures, weak convergence to a tight probability measure $\mu$ is equivalent to the pointwise convergence of the characteristic functions of $\mu\_n$ to the ch. f. of $\mu$. 
> 
> 
> 


This makes characteristic function(al)s more useful than in separable Banach spaces, where they are only used for uniqueness but not weak convergence.


