A fairly simple example is the Haar measure on $\mathbb{Q}_p$. We know that $\mathbb{Z}_p$ has measure $1$, and the measure is translation invariant, so it follow that $a+p\mathbb{Z}_p$ has measure $\frac{1}{p}$. We can do similarly for cosets of $p^n\mathbb{Z}_p$. See Chapter 2 of Cassels-Frohlich for details on this.

In this vain, one defines Haar measures on other number-theoretic objects, like adeles and ideles. Integration over these spaces can then be used to prove basic facts about more concrete objects, like zeta functions. For details, consult Koch's Number Theory, which gives many explicit examples of integration over $p$-adics and spaces of adeles and then uses them to prove Hecke's functional equation for the zeta function. (You can also find similar material in Cassels-Frohlich, though I find Koch to be much more readable.)

A fairly simple example is the Haar measure on $\mathbb{Q}_p$. If we scale the measure so that $\mathbb{Z}_p$ has measure $1$, and the measure is translation invariant, it follows that $a+p\mathbb{Z}_p$ has measure $\frac{1}{p}$. We can do similarly for cosets of $p^n\mathbb{Z}_p$. See Chapter 2 of Cassels-Frohlich for details on this.

In this vein, one defines Haar measures on other number-theoretic objects, like adeles and ideles. Integration over these spaces can then be used to prove basic facts about more concrete objects, like zeta functions. For details, consult Koch's Number Theory, which gives many explicit examples of integration over $p$-adics and spaces of adeles and then uses them to prove Hecke's functional equation for the zeta function. (You can also find similar material in Cassels-Frohlich, though I find Koch to be much more readable.)