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

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.