Properties of Fuzzy Measures
For any, a fuzzy measure is:
- additive if for all ;
- supermodular if ;
- submodular if ;
- superadditive if for all ;
- subadditive if for all ;
- symmetric if implies ;
- Boolean if or .
Understanding the properties of fuzzy measures is useful in application. When a fuzzy measure is used to define a function such as the Sugeno integral or Choquet integral, these properties will be crucial in understanding the function's behavior. For instance, the Choquet integral with respect to an additive fuzzy measure reduces to the Lebesgue integral. In discrete cases, a symmetric fuzzy measure will result in the ordered weighted averaging (OWA) operator. Submodular fuzzy measures result in convex functions, while supermodular fuzzy measures result in concave functions when used to define a Choquet integral.
Read more about this topic: Fuzzy Measure Theory
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