Example

— 1) Let $K = \mathbf{R}$. If $s > 0$ then $s \cdot [0,1] = [0,s]$; if $s < 0$, $s \cdot [0,1] = [s,0]$. Thus $\mathrm{mod}_\mathbf{R} t = |t|$ for all $t \in \mathbf{R}$.

2) Let $K = \mathbf{Q}_p$. If $s \in \mathbf{Q}_p^*$ is such that $|s|_p = p^{-n}$, then $s \mathbf{Z}_p$ is the set of $x \in \mathbf{Q}_p$ such that $|x|_p \leq p^{-n}$; therefore, if $\mu$ denotes the normalized Haar measure on $\mathbf{Q}_p$, then $\mu(s \mathbf{Z}_p) = p^{-n}$. Thus $\mathrm{mod}_{\mathbf{Q}_p} t = |t|_p$ for all $t \in \mathbf{Q}_p$.