Operational properties and matrix representations of quantum measures  

作　　者：GUO ZhiHua CAO HuaiXin CHEN ZhengLi YIN JunCheng 

机构地区：[1]College of Mathematics and Information Science, Shaanxi Normal University, Xi' an 710062, China

出　　处：《Chinese Science Bulletin》2011年第16期1671-1678,共8页

基　　金：supported by the National Natural Science Foundation of China (10871224, 10571113);the Natural Science Research Program of Shaanxi Province (2009JM1011)

摘　　要：Denoted by M(A),QM(A)and SQM(A)the sets of all measures,quantum measures and subadditive quantum measures on a σ-algebra A,respectively.We observe that these sets are all positive cones in the real vector space F(A)of all real-valued functions on A and prove that M(A)is a face of SQM(A).It is proved that the product of m grade-1 measures is a grade-m measure.By combining a matrix Mμto a quantum measureμon the power set An of an n-element set X,it is proved thatμν(resp. μ⊥ν)if and only if μν M M(resp.MμMv=0).Also,it is shown that two nontrivial measuresμandνare mutually absolutely continuous if and only ifμ·ν∈QM(An).Moreover,the matrices corresponding to quantum measures are characterized. Finally,convergence of a sequence of quantum measures on An is introduced and discussed;especially,the Vitali-Hahn-Saks theorem for quantum measures is proved.Denoted by M（A）, QM（A） and SQM（A） the sets of all measures, quantum measures and subadditive quantum measures on a σ-algebra A, respectively. We observe that these sets are all positive cones in the real vector space F（A） of all real-valued functions on A and prove that M（A） is a face of SQM（A）. It is proved that the product of m grade-1 measures is a grade-m measure. By combining a matrix Mμ to a quantum measureμ on the power set An of an n-element set X, it is proved that μ-〈-〈 v （resp. μ/ v ） if and only if M -〈-〈 Mv （resp. MμMv=0）. Also, it is shown that two nontrivial measures μ and v are mutually absolutely continuous if and only if μ. v∈ QM（An）. Moreover, the matrices corresponding to quantum measures are characterized. Finally, convergence of a sequence of quantum measures on An is introduced and discussed; especially, the Vitali-Hahn-Saks theorem for quantum measures is proved.

关 键 词：矩阵表示 量子 定理证明 性质 业务 质量管理 向量空间 实值函数 

分 类 号：O413[理学—理论物理]