Computing Haar Measures
Arno Pauly, Dongseong Seon, Martin Ziegler

TL;DR
This paper proves that Haar integrals on computably compact groups are computable, provides explicit algorithms with error bounds, and analyzes their complexity, linking them to classical integration and complexity classes.
Contribution
It establishes the computability of Haar integrals on computably compact groups and presents explicit algorithms with rigorous error analysis.
Findings
Haar integrals are computable on all computably compact groups.
Explicit algorithms with guaranteed convergence and error bounds are provided.
Haar integrals on SO(3) and SU(2) reduce to Euclidean integration and characterize #P$_1$.
Abstract
According to Haar's Theorem, every compact group admits a unique (regular, right and) left-invariant Borel probability measure . Let the Haar integral (of ) denote the functional integrating any continuous function with respect to . This generalizes, and recovers for the additive group , the usual Riemann integral: computable (cmp. Weihrauch 2000, Theorem 6.4.1), and of computational cost characterizing complexity class #P (cmp. Ko 1991, Theorem 5.32). We establish that in fact every computably compact computable metric group renders the Haar integral computable: once asserting computability using an elegant synthetic argument, exploiting uniqueness in a computably compact space of probability measures; and once presenting and analyzing an explicit, imperative algorithm based on…
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