Characterising the Haar measure on the $p$-adic rotation groups via inverse limits of measure spaces

We determine the Haar measure on the compact $p$-adic special orthogonal groups of rotations $\mathrm{S}\mathrm{O}(d{)}_p$ in dimension $d=2,3$, by exploiting the machinery of inverse limits of measure spaces, for every prime $p>2$. We characterise $\mathrm{S}\mathrm{O}(d{)}_p$ as inverse limits of finite groups, of which we provide parametrisations and orders, together with an equivalent description through a multivariable Hensel lifting. Supplying these finite groups with their normalised counting measures, we get an inverse family of Haar measure spaces for each $\mathrm{S}\mathrm{O}(d{)}_p$. Finally, we constructively prove the existence of the so-called inverse limit measure of these inverse families, which is explicitly computable, and prove that it gives the Haar measure on $\mathrm{S}\mathrm{O}(d{)}_p$. Our results pave the way towards the study of the irreducible projective unitary representations of the $p$-adic rotation groups, with potential applications to the recently proposed $p$-adic quantum information theory.