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.

翻译：我们建立了一个严格的测度论框架，用于分析$L^1(\mu)$空间中非扩张映射的不动点，并明确考虑了定点运算中产生的量化误差。我们的核心结果表明，在依测度局部收敛拓扑下紧致（我们称之为测度紧性）的$L^1(\mu)$的每个有界闭凸子集，对于非扩张映射都具有不动点性质。证明依赖于一致可积性、测度凸性以及正规结构理论中的技术，包括Kirk定理的应用。我们进一步通过将定点运算建模为非扩张映射的扰动来分析量化效应，在测度紧性条件下建立了近似不动点的存在性。我们还给出了反例，以说明我们假设条件的最优性。在理论发展之外，我们将此框架应用于一个人机协同编辑系统。通过将AI生成建议、人类编辑器和量化器之间的交互建模为测度紧集上非扩张映射的复合，我们证明了“稳定共识产物”的存在性。我们证明即使在有界量化误差下，这种共识状态仍保持为近似不动点，并给出了一个符合该框架的人机协同编辑循环的具体示例。我们的结果强调了测度论紧性在涉及人类与智能体的可靠协同系统设计与验证中的价值。