Computing haar integrals

Abstract

Define integral: $\mathcal{C}(X) \rightarrow \mathbb{R}$ to be a functional $f \mapsto \int_X f d\mu$ with a Borel probability measure $\mu$. Then, the usual Riemann integral on [0,1] is an integral on [0,1] with the Borel probability measure on the real line. This integral on [0,1] is known to be computable, and its complexity was analyzed. After that, more generally it is known that, on some spaces, an integral is computable if and only if its corresponding measure is computable. Consequently, to actually compute natural integrals on spaces, one should compute natural measures. Arguably, Haar measures can be seen as natural measures. This is because Haar's theorem states that for any compact topological group, there exists a unique Haar probability measure which is translation-invariant and regular. Thus, to compute natural integrals, we consider Haar measures to be natural and prove that these measures are computable. Another motivation to prove computability of Haar measures is that proving that Haar measures are computable can be interpreted as proving a computable version of Haar's theorem. In this paper, we prove that Haar integrals (integrals with their Haar measures) are computable under computable version of assumptions of Haar's theorem. Moreover, we prove computability, analyze complexity, and implement the Haar integral on arguably the most important compact topological group $\mathcal{SO}(3)$.