Is the Haar measure on $\mathbb{Q}_p$ complete? The field of $p$-adic numbers $\mathbb{Q}_p$ is locally compact, and so there exists a Haar measure on $(\mathbb{Q}_p,+)$. My question is whether a Haar measure on $\mathbb{Q}_p$ will also be a complete measure? The reason this question came to my mind is that I am studying about adeles and I noticed that we take the Haar measure on the infinite place, $\mathbb{R}$, to be the Lebesgue measure and not the Borel measure restricted to the Borel sets. So I wondered whether the Haar measure on $\mathbb{Q}_p$ is also complete, but I couldn't find any way to answer the question because we simply assert the existence of Haar measures on $\mathbb{Q}_p$ and move on to computing integrals with respect to this measure.

A Haar measure for a locally compact group $G$, like $\mathbb{Q}_p$, is not complete, because the Borel sigma algebra in the measure space $(G,B(G),\mu)$ is not complete.