Examples of an unbounded measurable subset of finite measure of the p-adic number field

Question

Let $\mathbb{Q}_p$ be the p-adic number field, $\mathbb{Z}_p$ its ring of integers. A subset of $\mathbb{Q}_p$ is called bounded if it is contained in a compact subset. Let $\mu$ be the Haar measure on $\mathbb{Q}_p$ such that $\mu(\mathbb{Z}_p) = 1$. I would like to know an example of an unbounded measurable subset whose measure is finite with respect to $\mu$.

Answer 1

An example of such a set is $$\bigcup_{n\ge 0} p^{-n} (1 + p^{2n+1} \mathbb{Z}_p).$$