Generalised Perspectives and Foundations in Non-Additive Measure Derivatives and Choquet Integration

Non-additive measures and integrals have become important mathematical tools in modern research. They offer powerful frameworks for modelling interaction, uncertainty, and handling incomplete data where traditional additive techniques fail. This paper focuses on the Choquet integral and the corresponding non-additive measure derivatives, which can be considered mutually inverse problems. A comprehensive review of existing direct and indirect approaches is provided for analysis of the derivatives, aiming to refine and generalise current results while highlighting their inconsistencies and limitations. In the discrete case, Choquet integration reduces to a finite sum, and the necessary and sufficient conditions for the existence of measure derivatives form a linear system. In the continuous case, three existing frameworks are evaluated: studying the invariance under rearrangement of integrating functions for the integral; using generalised integral equations for Choquet computations to derive formulas for the result of integration as well as for measure derivatives; and the resulting measure approach, which uses the inverse nature of integration and measure derivatives and pairs integrating and resulting measures to derive explicit derivative formulas. Ultimately, this work maps out current perspectives on non-additive measure derivatives with respect to the Choquet integral, and outlines possible directions for their future theoretical and applied advancements.