Let $(X, d)$ be a separable and locally compact metric space, $\mathcal{B}$ the Borel $\sigma$-algebra 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: $$\frac{1}{\tau_n} \int_0^{\tau_n} \mu(A-t) dt.$$

Can someone help me?

Thank you!

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: $$\frac{1}{\tau_n} \int_0^{\tau_n} \mu(A-t) dt.$$

Can someone help me?

Thank you!