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A distribution over $\mathbb{Z}_p$ is defined as an "additive" map $\mu$ s.t. $\mu(\mathbb{Z}_p)=\sum_{i=0}^{p-1}\mu(i+p\mathbb{Z}_p)$. But now if i want define a distribution over $\mathbb{Q}_p$ what is the " additivity" that this map should have? I think something like:

For every $x \in \mathbb{Q}_p$, for every $N \in\mathbb{Z} $ then $\mu(x+ p^N\mathbb{Z}_p)=\sum_{i=0}^{p-1}\mu(x+ip^N+p^{N+1}\mathbb{Z}_p)$

Thank you for suggestion and references!

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    $\begingroup$ Sure. And (since there is a basis of open sets that are compacts ?) distributions on the metric space $p^N\mathbb{Z}_p,|.|_p$ are finite measures $\endgroup$ – reuns Oct 4 '18 at 20:58

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