I am looking at p-adic distributions, and in this case p-adic measures. To say that $\mu$ is a distribution means that the arguments of $\mu$ are compact open subsets of $\mathbb{Z}_p$, $\mu$ is finitely additive, and the values $\mu$ takes are in $\mathbb{C}_p$. To say that $\mu$ is a measure means that $\mu$ is a distribution and the values $\mu$ takes are bounded. Let $\mu$ be a measure.

Suppose that $lim_{n\to\infty}\mu(a+p^n\mathbb{Z}_p)=0$ for all $a\in\mathbb{Z}_p$. Does this imply that $\mu\equiv 0$?