关于可加模糊集函数的一种分解

Decomposition for Additive Fuzzy Set Functions

在经典测度论中,每个可加集函数可表示为两个测度之差,即将集函数表示为其上、下变差之差。本文将不交变差的定义推广到了模糊集上,运用三角基本模T∞和S∞算子表示模糊集合间的运算,T∞证明了有限可加模糊集函数在有界不交变差条件下的约当分解,推出了有界不交变差集函数的全不交变差的一种表示形式。

In the classical measure theory,every additive set function can be represented by the subtraction of its upper variation and lower variation.In this paper,the definition of disjoint variation is generalized to the fuzzy set functions. The fundamental t-norm T∞ and S∞ are considered as the operation of the fuzzy sets.A version of the Jordan decomposition theorem is proved for finitely T∞-additive functions with bounded disjoint variation.The form of the total disjoint variation for the functions with bounded disjoint variation is available.