Example

— Let $\mathbf{Q}_p$ be the $p$-adic field, the completion of $\mathbf{Q}$ for the $p$-adic absolute value $|x|_p = p^{-v_p(x)}$ (GT, IX, §3, No. 2, Example 3). The elements of $\mathbf{Q}_p$ are called $p$-*adic numbers*. We denote again by $|x|_p$ the continuous extension of the $p$-adic absolute value to $\mathbf{Q}_p$. One has

for $x, y$ in $\mathbf{Q}$ (loc. cit.), hence for $x, y$ in $\mathbf{Q}_p$; moreover, if $|y|_p < |x|_p$ then $|x + y|_p = |x|_p$, because $|x|_p = |(x + y) - y|_p \leq \sup(|x + y|_p, |y|_p)$. If $(x_n)$ is a sequence of points of $\mathbf{Q}_p$ tending to $x \in \mathbf{Q}_p^*$, then $|x - x_n|_p < |x|_p$ and $|x - x_n|_p < |x_n|_p$ for $n$ sufficiently large, therefore $|x|_p = |x_n|_p$. This proves that, for every $x \in \mathbf{Q}_p^*$, $|x|_p$ is a power of $p$.

Let $\mathbf{Z}_p$ be the closure of $\mathbf{Z}$ in $\mathbf{Q}_p$; this is a subring of $\mathbf{Q}_p$; its elements are called $p$-*adic integers*. One has $|x|_p \leq 1$ for every $x \in \mathbf{Z}_p$. Conversely, let $x$ be an element of $\mathbf{Q}_p$ such that $|x|_p \leq 1$, and let us show that $x \in \mathbf{Z}_p$; there exists a sequence $(x_n)$ of elements of $\mathbf{Q}$ tending to $x$, and $|x_n|_p \leq 1$ for $n$ sufficiently large by what we have seen above; it suffices to show that $x_n$ belongs to $\mathbf{Z}_p$ for $n$ sufficiently large; in other words, we are reduced to the case that $x \in \mathbf{Q}$; then $x = a/b$ with $b$ relatively prime to $p$; for every integer $n > 0$, there exist $b'_n \in \mathbf{Z}$ and $h_n \in \mathbf{Z}$ such that $bb'_n + h_n p^n = 1$, whence $x = \frac{abb'_n + ah_n p^n}{b} = ab'_n + \frac{ah_n p^n}{b}$ and $|x - ab'_n|_p \leq p^{-n}$, therefore $ab'_n$ tends to $x$.

From this, it follows that the closed ball with center 0 and radius $p^{-n}$, identical to the open ball with center 0 and radius $p^{-n+1}$, is $p^n \mathbf{Z}_p$. The topological space $\mathbf{Q}_p$ is therefore zero-dimensional, hence totally disconnected (GT, IX, §6, No. 4).

Let us show that the integers $0, 1, \ldots, p^n - 1$ constitute a system of representatives of $\mathbf{Z}_p$ modulo $p^n \mathbf{Z}_p$. First, $|k - k'|_p > p^{-n}$ for two such integers $k$ and $k'$, therefore the classes modulo $p^n \mathbf{Z}_p$ of these integers are distinct. On the other hand, let $x \in \mathbf{Z}_p$; there exists a $k \in \mathbf{Z}$ such that $|x - k|_p \leq p^{-n}$; adding a suitable multiple of $p^n$ to $k$, one can suppose that $k \in [0, p^n - 1]$, and $x$ is congruent to $k$ modulo $p^n \mathbf{Z}_p$. Whence our assertion. This shows that $\mathbf{Z}_p / p^n \mathbf{Z}_p$ is canonically isomorphic to $\mathbf{Z}/p^n \mathbf{Z}$. One sees, moreover, that $\mathbf{Z}_p$ is precompact, hence compact since it is complete. Since $\mathbf{Z}_p$ is an open subgroup of $\mathbf{Q}_p$, $\mathbf{Q}_p$ is locally compact. The topology of $\mathbf{Q}_p$ has a countable base (GT, IX, §2, No. 9, Cor. of Prop. 16). The additive group $\mathbf{Q}_p$ may be identified with the inverse limit of the discrete groups $\mathbf{Q}_p / p^n \mathbf{Z}_p$.

There exists one and only one Haar measure $\alpha$ on the additive group $\mathbf{Q}_p$ such that $\alpha(\mathbf{Z}_p) = 1$; it is called the normalized Haar measure on $\mathbf{Q}_p$. Since $\mathbf{Z}_p$ is the union of $p^n$ disjoint cosets of $p^n \mathbf{Z}_p$ ($n$ an integer $\geq 0$), one has $\alpha(p^n \mathbf{Z}_p) = p^{-n}$; similarly $\alpha(p^{-n} \mathbf{Z}_p) = p^n$, so that, finally, $\alpha(p^n \mathbf{Z}_p) = p^{-n}$ for every $n \in \mathbf{Z}$. By Prop. 8 b), $\alpha$ is the only positive measure on $\mathbf{Q}_p$ such that every coset of $p^n \mathbf{Z}_p$ (n an integer $\geq 0$) has measure $p^{-n}$.

The restriction of $\alpha$ to $\mathbf{Z}_p$ is obviously a Haar measure on $\mathbf{Z}_p$.