This is partially motivated by a question I saw earlier here, Does such a finitely additive function exist?
I've been reading about the topology of $\mathbb{Q}_p$ in Knapp's Advanced Algebra in Chapter 6, and I'm wondering if it's possible to impose a finitely additive function on $\mathbb{Q}_p$ in a natural way. By this I mean, suppose I take closed balls of form $ B(x,r)=\{y\in\mathbb{Q}_p\mid|x-y|\leq p^{-r}\} $ and let $\mathscr{B}$ be the set ring consisting of finite unions of such balls.
I suspect there must also be a finitely additive function $\mu\colon\mathscr{B}\to\mathbb{R}^{+}$, the nonnegative reals, such that $\mu(B(x,r))=p^{-r}$, as one would expect naturally?
In the case for $\mathbb{Q}$, such a function was constructed by first constructing a function which mapped a member of the set ring to its intersection with $\mathbb{Q}$. However, I view elements of $\mathbb{Q}_p$ as sequences in $\prod_{j=1}^\infty\mathbb{Q}$, so would their be some similar way of constructing such a function by first considering finite unions of such closed balls of sequences in $\prod_{j=1}^\infty\mathbb{R}$?
And unlike the case in $\mathbb{Q}$, would it in fact be possible to extend $\mu$ to a unique measure on the generated $\sigma$-algebra of $\mathscr{B}$?
Edit: Harry Altman's answer shows that the Haar measure is a measure which restricts back to this property. I guess I'm curious on how this would be built up. I mean, what would be the basic finitely additive function on $\mathscr{B}$ that could be uniquely extended to the Haar measure on the generated $\sigma$-algebra?
Thank you for your considerations.