Q:

Is the following sequence convergent in the weak topology?

Consider the metric space $X = \mathbb{R}$, $\mathcal{B}$ the Borel $\sigma$-algebra on $\mathbb{R}$ and $\mu$ a probability measure on $X$. Let $A \in \mathcal{B}$ and $\tau_n \nearrow \infty$ a sequence of positive numbers. 
I can't demonstrate that the following sequence of measures is or is not convergent in the weak topology:
$$\mu_n\left(A\right)=\frac{1}{\tau_n} \int_0^{\tau_n} \mu(A-t) dt.$$
Can someone help me?
Thank you!

A:

Observe that for any $x\in\mathbb R$ and $\delta\gt 0$, the inclusion 
$$\left(-\infty,x-t\right]\subset\left(-\infty,x-\delta\tau_n\right]   $$
holds for $\tau_n\delta\leqslant t\leqslant \tau_n$. Therefore, 
$$\mu_n\left(-\infty,x\right]\leqslant \delta +\mu\left(-\infty,x-\delta\tau_n\right]$$
and it follows that $\mu_n\left(-\infty,x\right]\to 0$ as $n$ goes to infinity. 

