Convergence in distrubution sense and convergence of distribution to a distrubution. The convergence in the test function is as follow :


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*Let $\mathcal D(U)$ the space of test function. $(\varphi _n)$ to $\varphi $ in $\mathcal D(U)$ if there is a compact $K\subset U$ that contain all support of the $\varphi _k$'s and $\partial ^\alpha \varphi _k\to \partial ^\alpha \varphi $ uniformly for all multi index $\alpha $.

*Let $T\in \mathcal D'(U)$ a distribution. Now, if $(T_k)$ is a sequence of $\mathcal D'(U)$, it convergence to $T\in D'(U)$ if $$\lim_{n\to \infty }\left<T_n,\varphi \right>=\left<T,\varphi \right>.$$
This looks to be a sort of "pointwise convergence" definition. 
Question 1
Is this the convergence in distribution sense ? 
Question 2
Let for example $$f_n(x)=\frac{n}{\sqrt\pi }e^{-n^2x^2}. $$
It is such that $$\lim_{n\to \infty }\left<f_n,\varphi \right>=\lim_{n\to \infty }\int_{\mathbb R}\varphi f_n(x)\,\mathrm d x=\left<\delta ,\varphi \right>.$$
So what is the situation here ? We can define an absolute continuous measure $\mu_n(dx)=f_n(x)dx$. Now, the distribution $\mu_n$ converge in distribution sense to the dirac distribution $\delta $ ? (Well, in probability $\mu_n$ define a probability measure, and this is how is define the convergence in law. Is there a link between these convergences ?)  
 A: Question 1. Yes, this is often taken as the definition. The space $\mathscr D'$ of distributions is in fact the dual of the space $\mathscr D$ of test functions where $\mathscr D$ is endowed with the finest locally convex topology making all inclusions $\mathscr D(K)\hookrightarrow \mathscr D$ for compact sets $K$ continuous where $\mathscr D(K)$ is the space of smooth functions with support in $K$ endowed with the topology of uniform convergence of all derivatives. $\mathscr D$ is thus the locally convex inductive limit of Frechet spaces, a so-called LF-space. Then the pointwise convergence of distributions is precisely the convergence in the weak$*$ topology $\sigma(\mathscr D',\mathscr D)$. The weak$*$ topology is convenient for several purposes but it also has certain disadvantages. For example it is not complete and it does not have any of the good locally convex properties like barrelledness or ultrabornologicity which allow direct applications of fnctional analytic principles like the Banach-Steinhaus or the open mapping theorem. In this respect, the strong topology $\beta(\mathscr D',\mathscr D)$ of uniform convergence on bounded subsets of $\mathscr D$ (which, by a result of Dieudonne and Schwartz, are already bounded in some $\mathscr D(K)$) is much better.
Question 2. What probabilists call weak convergence of probability measures is the pointwise convergence $\int \varphi d\mu_n$ for all bounded continuous functions. 
Since there are much more continuous functions than smooth ones with compact support, this is a priori stronger than convergence in $\mathscr D'$. However, either by locally convex theory (the Banach-Alaoglu theorem) or by approximating continuous $\varphi$ by smooth and compactly supported ones (which can be easily done uniformly on compact sets -- you then need that your sequence $\mu_n$ "essentially lives on a compact set" which is often called tightness) both types of convergence often coincide.
