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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Guo, Z., Cao, H., Chen, Z. et al. Operational properties and matrix representations of quantum measures. Chin. Sci. Bull. 56, 1671–1678 (2011). https://doi.org/10.1007/s11434-011-4481-4
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DOI: https://doi.org/10.1007/s11434-011-4481-4