Non-discrete \(k\)-order additivity of a set function and distorted measure. (English) Zbl 1522.28026
Summary: In this study, we generalize the concept of the \(k\)-order additivity of a set function. First, we discuss the Möbius transform for a non-discrete set function. Next, we generalize the definition of the \(k\)-order additivity of a set function using the Möbius transform and provide the equivalent conditions for the \(k\)-order additivity. Furthermore, we consider the \(k\)-order additivity of the distorted monotone measure. We prove that under certain conditions, a distorted measure is \(k\)-order additive if and only if the distortion function is a polynomial of \(k\)-th order.
Keywords:
fuzzy measure; monotone measure; \(k\)-order additivity; Möbius transform; distorted measureReferences:
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