# Distribution over $\mathbb{Q}_p$

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!

• 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 – reuns Oct 4 '18 at 20:58