Example of weak convergence I'd like to understand better the concept of weak convergence.
I know that a sequence of probability measures $\mu_n$ converges weakly to $\mu$ if $\int{f d\mu_n}$ converges to $\int{f d\mu}$ for each $f$ which is continuous and bounded. 
Could you please give me an example of a sequence of probability measures $\mu_n$ that converges weakly to $\mu$ and find a function $f$ such that $\int{f d\mu_n}$ does not converge to $\int{f d\mu}$?

Take on $[0,1]$ the measures $\mu_n=\frac{1}{n}\sum_{k=1}^n\delta_{k/n}$.
They converge weakly to the Lebesgue measure $\mu=\lambda^1$ restricted to $[0,1]\,$, because the Riemann sum $\mu_n(f)=\frac{1}{n}\sum_{k=1}^n f(\frac{k}{n})$ converges to its integral $\mu(f)=\int_{[0,1]}f(x)\,dx$ for every bounded continuous function $f$.
Let's break the continuous condition, but be sure not to take a Lipschitz function (see Portmanteau theorem https://en.wikipedia.org/wiki/Convergence_of_measures)
Let $f$ be the Dirichlet function $f=\chi_{\mathbb{Q}\cap[0,1]}$.
You have $1=\mu_n(f)\neq\mu(f)=0\quad\forall n\in\mathbb{N}$
