Is it possible to do calculus problems over the $p$-adic numbers, $\mathbb{Q}_p$ ? Let $d\mu = \frac{dx}{|x|_p}$ be the Haar measure on $\mathbb{Q}_p^\times$. What would be the value of $$ \int_{\mathbb{Q}_p} |x|_p^2 \,d\mu $$ Does this question even make sense. Let's look at the corresponding question over $\mathbb{R}$. Then $G = \mathbb{R}^\times$ and the Haar measure is $\frac{dx}{x}$.
- $(\mathbb{R}, dx)$
- $(\mathbb{R}^\times, \frac{dx}{|x|})$
So those are two examples of Haar measures over locally compact groups. For any measureable set $A \subset \mathbb{Q}_p$ we have:
- $\mu(\mathbb{Z}_p^\times) = 1$
- $\mu(xA) = \mu(A)$ (traslation invariant)
- $\mathbb{Q}_p^\times= \bigcup_{k \in \mathbb{Z}} \; p^k \mathbb{Z}_p^\times$
In the $p$-adic numbers there are certain integrals that can be defined. The book I'm reading as a spoiler: $$ \int_{\mathbb{Z}_p \backslash \{ 0\}} |x|^2_p \, d^\times x = \frac{1}{1-p^2} $$ which he argues using the geometric series. The only integrable function I could think of is $x^2$, maybe other arithemtic functions are integrable, as well.