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On the strong regularity with the multifractal measures in a probability space

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Abstract

We prove a decomposition theorem of Besicovitch’s type for the relative multifractal Hausdorff measure and packing measure in a probability space. By obtaining a new necessary condition for the strong regularity with the multifractal measures in a more general framework, we extend in this paper the density theorem of Dai and Li (A multifractal formalism in a probability space. Chaos Solitons Fractals 27:57–73, 2006). In particular, this result is more refined than those found in Dai and Taylor (Defining fractal in a probability space. Ill J Math 38:480–500, 1994).

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Acknowledgements

The author would like to thank Najmeddine Attia for his interesting comments and many valuable suggestions on this work.

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  1. Department of Mathematics, Faculty of Sciences of Monastir, University of Monastir, 5019, Monastir, Tunisia

    Bilel Selmi

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  1. Bilel Selmi
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Selmi, B. On the strong regularity with the multifractal measures in a probability space. Anal.Math.Phys. 9, 1525–1534 (2019). https://doi.org/10.1007/s13324-018-0261-5

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