Quantization Errors, Human--AI Interaction, and Approximate Fixed Points in $L^1(\mu)$

Abstract

We develop a rigorous measure-theoretic framework for the analysis of fixed points of nonexpansive maps in the space $L^1(\mu)$, with explicit consideration of quantization errors arising in fixed-point arithmetic. Our central result shows that every bounded, closed, convex subset of $L^1(\mu)$ that is compact in the topology of local convergence in measure (a property we refer to as measure-compactness) enjoys the fixed point property for nonexpansive mappings. The proof relies on techniques from uniform integrability, convexity in measure, and normal structure theory, including an application of Kirk's theorem. We further analyze the effect of quantization by modeling fixed-point arithmetic as a perturbation of a nonexpansive map, establishing the existence of approximate fixed points under measure-compactness conditions. We also present counterexamples that illustrate the optimality of our assumptions. Beyond the theoretical development, we apply this framework to a human-in-the-loop co-editing system. By formulating the interaction between an AI-generated proposal, a human editor, and a quantizer as a composition of nonexpansive maps on a measure-compact set, we demonstrate the existence of a "stable consensus artefact". We prove that such a consensus state remains an approximate fixed point even under bounded quantization errors, and we provide a concrete example of a human-AI editing loop that fits this framework. Our results underscore the value of measure-theoretic compactness in the design and verification of reliable collaborative systems involving humans and artificial agents.