What are the null set in $\mathbb{Z}_p$?

Question

I was wondering does all null set in $\mathbb{Z}_p$ (wtr to the Haar measure) are countable?

This is not true for things in real setup (like $\mathbb{R}$ or $S^1$) but maybe for p-adic things it is true.

Thank

Answer 1

Would the following fit the bill in the case $p>2$?

Let

$$S=\{\sum_{n=0}^\infty a_ip^i\in\Bbb{Z}_p\mid \text{$a_i\in\{0,1\}$ for all $i$}\}.$$

For any natural number $m$ the elements of $S$ are constrained to have one of $2^m$ initial sequences of $p$-adic digits, each sequence defining a coset of $p^m\Bbb{Z}_p$. Hence the measure of $S$ is bounded from above by $(2/p)^m$ for all $m$. In other words, $S$ is a null set. Obviously $S$ is uncountable.