局部紧Hausdorff空间上的Fuzzy测度

Fuzzy measure on locally compact Hausdorff space

研究局部紧Hausdorff空间X上的Fuzzy测度的正则性。首先引入X上内(外)正则集,正则集以及正则Fuzzy测度的概念,并给出了Fuzzy测度正则的充要条件和任意两个紧(或紧Gδ)集的正常差内(外)正则的条件;其次证明单调递增的内正则集的并是内正则的,具有有限Fuzzy测度的单调递减的外正则集的交是外正则的;最后在严格单调条件下,证明具有有限Fuzzy测度的有限个两两不交内正则集的并是内正则的以及每一个紧(或紧Gδ)集是外正则的当且仅当每一个有界开集是内正则的。

The regularity of fuzzy measure on locally compact Hausdorff space X is studied. Firstly, the concepts of the inner （outer） regular set, regular set and regular fuzzy measure on X are given. A necessary and sufficient condition under which a fuzzy measure is regular is obtained. At same time, the sufficient conditions under which every proper difference of two compact （ or compact Gδ） sets is inner （outer） regular are shown. Secondly, it is shown that the union of an increasing sequence of inner regular sets is inner regular, and the intersection of outer regular sets with finite measure is outer regular. Finally, for strictly monotone fuzzy measure, it is proved that the finite disjoint union of inner regular sets with finite measure is inner regular and every compact （ or compact Gδ ） set is outer regular, if and only if every bounded open set be inner regular.