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Characterization of a coherent upper conditional prevision as the Choquet integral with respect to its associated Hausdorff outer measure

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  • Serena Doria

Abstract

A model of coherent upper conditional prevision for bounded random variables is proposed in a metric space. It is defined by the Choquet integral with respect to Hausdorff outer measure if the conditioning event has positive and finite Hausdorff outer measure in its Hausdorff dimension. Otherwise, when the conditioning event has Hausdorff outer measure equal to zero or infinity in its Hausdorff dimension, it is defined by a 0–1 valued finitely, but not countably, additive probability. If the conditioning event has positive and finite Hausdorff outer measure in its Hausdorff dimension it is proven that a coherent upper conditional prevision is uniquely represented by the Choquet integral with respect to the upper conditional probability defined by Hausdorff outer measure if and only if it is monotone, comonotonically additive, submodular and continuous from below. Copyright Springer Science+Business Media, LLC 2012

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  • Serena Doria, 2012. "Characterization of a coherent upper conditional prevision as the Choquet integral with respect to its associated Hausdorff outer measure," Annals of Operations Research, Springer, vol. 195(1), pages 33-48, May.
    • Handle: RePEc:spr:annopr:v:195:y:2012:i:1:p:33-48:10.1007/s10479-011-0899-y
    DOI: 10.1007/s10479-011-0899-y
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    References listed on IDEAS

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    1. Schmeidler, David, 1989. "Subjective Probability and Expected Utility without Additivity," Econometrica, Econometric Society, vol. 57(3), pages 571-587, May.
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    1. Serena Doria, 2015. "Symmetric coherent upper conditional prevision defined by the Choquet integral with respect to Hausdorff outer measure," Annals of Operations Research, Springer, vol. 229(1), pages 377-396, June.
    2. Serena Doria, 2022. "Coherent Upper Conditional Previsions Defined through Conditional Aggregation Operators," Mathematics, MDPI, vol. 10(24), pages 1-13, December.
    3. Serena Doria, 2019. "Preference orderings represented by coherent upper and lower conditional previsions," Theory and Decision, Springer, vol. 87(2), pages 233-252, September.
    4. Serena Doria, 2017. "On the disintegration property of coherent upper conditional prevision defined by the Choquet integral with respect to its associated Hausdorff outer measure," Annals of Operations Research, Springer, vol. 256(2), pages 253-269, September.
    5. Serena Doria & Radko Mesiar & Adam Šeliga, 2020. "Sub-Additive Aggregation Functions and Their Applications in Construction of Coherent Upper Previsions," Mathematics, MDPI, vol. 9(1), pages 1-12, December.

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