Computing Haar Measures

According to Haar's Theorem, every compact group $G$ admits a unique left-invariant Borel probability measure $\mu_G$. Let the Haar integral (of $G$) denote the functional $\int_G:\mathcal{C}(G)\ni f\mapsto \int f\,d\mu_G$ integrating any continuous function $f:G\to\mathbb{R}$ with respect to $\mu_G$. This generalizes, and recovers for the additive group $G=[0;1)\mod 1$, the usual Riemann integral: computable (cmp. Weihrauch 2000, Theorem~6.4.1), and of computational cost characterizing complexity class #P$_1$ (cmp. Ko 1991, Theorem~5.32). We establish that in fact every computably compact computable metric group renders the Haar integral computable: once using an elegant synthetic argument; and once presenting and analyzing an explicit, imperative algorithm. Regarding computational complexity, for the groups $\mathcal{SO}(3)$ and $\mathcal{SU}(2)$ we reduce the Haar integral to and from Euclidean/Riemann integration. In particular both also characterize #P$_1$.