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Generalized fractal dimensions and gauges for self-similar sets and their application in the assessment of coherent conditional previsions and in the calculation of the Sugeno integral

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  • Achour, Rim
  • Doria, Serena
  • Selmi, Bilel
  • Li, Zhiming

Abstract

In this paper, we compute the generalized Hausdorff and packing dimensions of self-similar sets that meet the open set condition. We thoroughly characterize the class of Hausdorff gauges and generalized pre-packing gauges for a self-similar set A that satisfies the open set condition under certain criteria. We derive a more general necessary and sufficient condition for a gauge function to be a Hausdorff gauge for a set A, packing gauge, or pre-packing gauge. Additionally, we estimate the associated Hausdorff measures and packing pre-measures. Finally, we apply these results to assess coherent conditional provisions and to calculate the Sugeno integral with respect to the Lebesgue measure, of the generalized Hausdorff measures of some self-similar sets, such as the middle third Cantor set, the Sierpinsky carpet, and the Sierpinsky triangle.

Suggested Citation

  • Achour, Rim & Doria, Serena & Selmi, Bilel & Li, Zhiming, 2025. "Generalized fractal dimensions and gauges for self-similar sets and their application in the assessment of coherent conditional previsions and in the calculation of the Sugeno integral," Chaos, Solitons & Fractals, Elsevier, vol. 196(C).
    • Handle: RePEc:eee:chsofr:v:196:y:2025:i:c:s096007792500387x
    DOI: 10.1016/j.chaos.2025.116374
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    References listed on IDEAS

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    1. 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.
    2. Doria, Serena, 2024. "Merging of coherent upper conditional probabilities defined by Hausdorff outer measures," Chaos, Solitons & Fractals, Elsevier, vol. 181(C).
    3. Serena Doria, 2022. "Coherent Upper Conditional Previsions Defined through Conditional Aggregation Operators," Mathematics, MDPI, vol. 10(24), pages 1-13, December.
    4. Achour, Rim & Li, Zhiming & Selmi, Bilel & Wang, Tingting, 2024. "A multifractal formalism for new general fractal measures," Chaos, Solitons & Fractals, Elsevier, vol. 181(C).
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