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Coherent Upper and Lower Conditional Previsions Defined by Hausdorff Outer and Inner Measures

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Modeling, Design, and Simulation of Systems with Uncertainties

Part of the book series: Mathematical Engineering ((MATHENGIN,volume 3))

Abstract

A new model of coherent upper conditional previsions is proposed to represent uncertainty and to make previsions in complex systems. 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.Moreover sufficient conditions are given such that the upper conditional previsions satisfy the disintegration property and the conglomerability principle.

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Authors and Affiliations

  1. Department of Sciences, University G. D’Annunzio, Chieti-Pescara, Italy

    Serena Doria

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  1. Serena Doria
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Correspondence to Serena Doria .

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Editors and Affiliations

  1. , Chair of Mechatronics, University of Rostock, Justus-von-Liebig-Str. 6, Rostock, 18059, Germany

    Andreas Rauh

  2. , Faculty of Engineering, INKO, University of Duisburg-Essen, Duisburg, 47048, Germany

    Ekaterina Auer

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Doria, S. (2011). Coherent Upper and Lower Conditional Previsions Defined by Hausdorff Outer and Inner Measures. In: Rauh, A., Auer, E. (eds) Modeling, Design, and Simulation of Systems with Uncertainties. Mathematical Engineering, vol 3. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15956-5_9

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  • DOI: https://doi.org/10.1007/978-3-642-15956-5_9

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  • Online ISBN: 978-3-642-15956-5

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