一类半连续模糊测度的扩张

模糊测度的扩张是模糊测度论中一个非常重要和难度较大的课题。本文对1]提出的SC-Fuzzy测度证明了如下扩张定理。 定理 设IR是具有有限覆盖性的一个代数,由|R产生的σ(|R)具有,对任何A,B∈σ(|R),AB且A≠B,{{X_m};{X_M}|R,X_m↑∪ X_m A,对任何{Y_m} m=1|R,Y_m↑∪ Y_m B总有∪ X_m ∪Y_m}≠φ,μ是|R上的SC-Fuzzy测度 m=1 m=1 m=1则μ可以唯一地扩张到σ(|R)上去。

The Extensions of A Class of SC-Fuzzy Measures

The Extension of fuzzy measures is a very important and rather difficult problem in fuzzy measure theory. In this paper, we prove the following extension theorem. Theorem. Suppose that |R is a field, with finite covering property. If δ(| R) satisfies the condition that then any SC-Fuzzy Mealure which is defined on |R can be extended uniguely to δ(|R).