Non-Additive Measure Theory and Fuzzy Integrals
Summary
Non-additive measure theory extends classical measure by relaxing the requirement that the measure of a union of disjoint sets equals the sum of their measures. Such set functions, often called fuzzy measures or capacities, allow interactions among events and capture phenomena such as ambiguity, synergy and redundancy. Fuzzy integrals—most notably the Choquet and Sugeno integrals—generalise the Lebesgue integral to non-additive measures, providing tools for aggregation that respect the underlying interactions. The Choquet integral interpolates between additive expectation and extreme‐value criteria, making it suitable for decision making under uncertainty, risk assessment and sensitivity analysis. The Sugeno integral, based on sup–min operators, excels in ordinal settings such as qualitative decision models and image processing. Mathematical foundations draw on Möbius transforms to represent capacities, translation and homogeneity properties to analyse integral behaviour, and novel time‐scale formulations to unify discrete, continuous and quantum calculi. Recent theoretical advances have addressed convergence theorems, optimal transport distances for capacities and fractional‐order operators on non‐additive measure spaces. Applications span economics, artificial intelligence, machine learning, multicriteria decision support, and environmental modelling, where non‐additive integrals accommodate ambiguity and nonlinear interactions beyond the reach of classical probability and expectation theories.
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Non-Additive Measure Theory and Fuzzy Integrals publication trend
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Technical terms
Non-additive measure: A monotone set function that does not require additivity for disjoint sets, enabling the modelling of interactions and ambiguity.
Fuzzy measure (capacity): A generalisation of a probability measure defined on all subsets of a universe, characterised by monotonicity but not necessarily additivity.
Choquet integral: An expectation‐type integral with respect to a non-additive measure, aggregating both marginal contributions and interactions among events.
Sugeno integral: An idempotent aggregation operator for fuzzy measures based on sup–min composition, suitable for ordinal and qualitative data.
Möbius transform: A representation of a capacity as a signed measure on all subsets, facilitating computation of fuzzy integrals and interaction indices.
References
- The transport problem for non-additive measures. European Journal of Operational Research (2023).
- Merging of coherent upper conditional probabilities defined by Hausdorff outer measures. Chaos Solitons & Fractals (2024).
- Coherent lower and upper conditional previsions defined by Hausdorff inner and outer measures to represent the role of conscious and unconscious thought in human decision making. Annals of Mathematics and Artificial Intelligence (2021).
- Δ -Choquet integral on time scales with applications. Chaos Solitons & Fractals (2022).
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