Operational properties and matrix representations of quantum measures

• 来源期刊: 科学通报(英文版)
• 学科: 物理学
• 年，卷（期）: 2011,（16）
• 页码范围: 1671-1678
• 页数: 8页
• 分类号: O413
• 语种: 中文

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.