Computing Haar Measures

Abstract

According to Haar's Theorem, every compact group G admits a unique left-invariant Borel probability measure µG. Let the Haar integral (of G) denote the functional G : C(G) f → f dµG integrating any continuous function f : G → R with respect to µG. 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 #P1 (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 SO(3) and SU(2) we reduce the Haar integral to and from Euclidean/Riemann integration. In particular both also characterize #P1. ACM Subject ClassificationTheory of computation → Constructive mathematics; Mathematics of computing → Continuous mathematics Keywords and phrases computable analysis, topological groups, exact real arithmetic, Haar measure Digital Object Identifier 10.4230/LIPIcs.CSL.2020.00 European Union's Horizon 2020 MSCA IRSES project 731143. Motivation and OverviewComplementing empirical approaches, heuristics, and recipes [23,25], Computable Analysis[30] provides a rigorous algorithmic foundation to Numerics, as well as a way of formally measuring the constructive contents of theorems in classical Calculus. Haar's Theorem is such an example, of particular beauty combining three categories: compact metric spaces, algebraic groups, and measure spaces:Fact 1. Let (G, e, •, • −1 ) denote a group and (G, d) a compact metric space such that the group operation • and inverse operation • −1 are continuous with respect to d (that is, a topological group). There exists a unique left-invariant Borel probability measure µ G , called Haar measure, on G.We refrain from expanding on generalizations to locally-compact Hausdorff spaces. Recall that a left-invariant measure satisfies µ(U ) = µ(g • U ) for every g ∈ G and every measurable