Haar-Type Measures on Topological Quasigroups and Kunen's Theorem

TL;DR

This paper extends the concept of Haar measures to topological quasigroups, introducing quasi-invariant measures with modular cocycles, and explores how Moufang identities influence these measures, linking algebraic identities to measure-theoretic properties.

Contribution

It develops a framework for Haar-type measures on topological quasigroups and analyzes the impact of Moufang identities on modular cocycles, connecting algebraic and measure-theoretic structures.

Findings

Quasi-invariant measures with modular cocycles are suitable for topological quasigroups.

Moufang identities impose strong constraints on the modular cocycle.

Kunen's theorem relates loop structures to the collapse of modular defects.

Abstract

Haar measure is a fundamental structure in harmonic analysis on locally compact groups. Its existence reflects the compatibility between topology and the associative algebraic structure of groups. In this paper we propose a framework for Haar-type measures on topological quasigroups. Since associativity is absent, strict translation invariance is generally too strong to expect. We therefore introduce quasi-invariant measures whose defect is measured by a modular cocycle attached to translations. We then explain, in a detailed and cautious form, how Moufang-type identities may impose strong constraints on this cocycle. In particular, under additional quasi-invariance assumptions for right translations, the Moufang-type identity $(N1)$ leads naturally to a multiplicativity relation for the cocycle. This suggests a measure-theoretic interpretation of Kunen's theorem: the emergence of loop structure may be viewed as the collapse of a modular defect in the translation geometry of a quasigroup.