Operational properties and matrix representations of quantum measures |
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Authors: | ZhiHua Guo HuaiXin Cao ZhengLi Chen JunCheng Yin |
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Institution: | College of Mathematics and Information Science, Shaanxi Normal University, Xi' an 710062, China |
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Abstract: | 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 M
μ to a quantum measure μ on the power set A
n
of an n-element set X, it is proved that μ ≪ ν (resp. μ ⊥ ν) if and only if M
μ ≪ M
ν
(resp. M
μ
M
ν
=0). Also, it is shown that two nontrivial measures μ and ν are mutually absolutely continuous if and only if μ·ν∈QM(A
n
). Moreover, the matrices corresponding to quantum measures are characterized. Finally, convergence of a sequence of quantum
measures on A
n
is introduced and discussed; especially, the Vitali-Hahn-Saks theorem for quantum measures is proved. |
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Keywords: | quantum measure absolute continuity product matrix representation convergence |
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