Let $f:\mathbb Z^d \rightarrow \mathbb Z^d$ be a linear map having determinant 1. Is there an obvious way to see that if $U\subseteq \mathbb Z_p^d$ is a measurable set, then the p-adic measure of $U$ is equal to the p-adic measure of $f(U)$? I mean, without involving change of coordinates and jacobians.
Many thanks!