Abstract
In this note, it is proved that Liu’s uncertain measure and Choquet capacity differ from each other. Specifically, over any locally compact separable metric space such as \(\mathbb {R}^n\), a Liu’s uncertain measure is generally not a Choquet capacity. Further, Liu’s uncertain measure is not an ordinary probability measure in the sense of Kolmogorov. Notice that Choquet theory is consistent with Kolmogorov’s probability theory, as a direct generalization of the latter by allowing the random elements to take closed sets of the underlying space as the values.
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Acknowledgements
This work was supported by National Natural Science Foundation of China (NSFC) (11871394, 71871121), and HRSA, US Department of Health & Human Services (No. H49MC0068).
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Wei, G., Li, Z., Cai, M. (2021). A Note on the Relation Between Liu’s Uncertain Measure and Choquet Capacity. In: Kreinovich, V. (eds) Statistical and Fuzzy Approaches to Data Processing, with Applications to Econometrics and Other Areas. Studies in Computational Intelligence, vol 892. Springer, Cham. https://doi.org/10.1007/978-3-030-45619-1_18
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