# Can this function extend to a measure?

I believe I have one final question on a function I've been thinking about.

To set up, let $\mathbb{Q}_p$ be the $p$-adic numbers, and let $B(x,r)$ denote the closed balls $$B(x,r)=\{y\in\mathbb{Q}_p:|x-y|\leq p^{-r}\}$$ and $\mathscr{B}$ be the set ring of finite unions of such closed balls. Since $\mathbb{Q}_p$ is an ultrametric space, then any two balls are either disjoint, or one is contained in the other. So there is a finitely additive function $\mu$ such that $\mu(B(x,r))=p^{-r}$, since for any finite union one can sum of the maximal balls in the union.

Does this function $\mu$ extend to a unique measure on the $\sigma$-algebra generated by $\mathscr{B}$? I believe by Caratheodory's theorem, if $\mu$ is countably additive, then it follows in the affirmative that $\mu$ does extend to a unique measure. If this is correct, is there an easy way to see why $\mu$ would be countably additive? Thanks.

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Since $p^{-r}$ decreases as $r$ increases, at the most you will get a "signed measure", i.e., an $\mathbb{R}$-valued measure. Is that what you want? – Pete L. Clark Nov 6 '11 at 0:36
@Pete Yes, Professor Clark, I believe so. Is there a relatively elementary explanation as to how $\mu$ extends to such a measure? – Buble Nov 6 '11 at 1:49