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.

中文翻译：

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集合函数的非离散 k 阶可加性和失真测度

在这项研究中，我们推广了集合函数的k阶可加性的概念。首先，我们讨论非离散集函数的莫比乌斯变换。接下来，我们使用莫比乌斯变换概括了集合函数的k阶可加性的定义，并提供了k阶可加性的等价条件。此外，我们考虑了扭曲单调测度的k阶可加性。我们证明，在某些条件下，当且仅当失真函数是 k 阶多项式时，失真度量是 k 阶加法的。