Q:

Gaussian distributions - a question about convergence

Let $\mu_n$ be Gaussian distributions with mean $0$ and standard deviation $1/n$ and $f$ a function.
It may be true that if $\underset{\mathbb{R}}{\int} f \mu_n dx \rightarrow  \underset{\mathbb{R}}{\int} f \delta_0dx$, then $\mu_n \rightarrow \delta_0$? Why yes or not?  If yes, in what conditions? 
$\underset{\mathbb{R}}{\int} f \mu_n dx \rightarrow  \underset{\mathbb{R}}{\int} f \delta_0dx$ $\Leftrightarrow \underset{\mathbb{R}}{\int}f (\mu_n - \delta_0) dx \rightarrow 0 \Leftrightarrow \mu_n \rightarrow \delta_0$ ?
Thank you!

A:

Let $\mu_n$ be the gaussian distribution with mean $0$ and standard deviation $1/n$. 
Then $\displaystyle\int_\mathbb R f(x) \mu_n(\mathrm dx) \to f(0)$ for every bounded measurable function $f$ continuous at $0$.

