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Generating a Probability Measure on the Completion of a Fractal Structure

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Abstract

Fractal structures were introduced to characterize non-archimedean quasi-metrization and they can be used to study self similar sets and fractals in general. In this paper we show how to construct a probability measure by defining it iteratively on some key subsets. In fact, given a space with a fractal structure and a pre-measure on some families of subsets determined by the fractal structure, we prove that it can be extended to a measure on the Borel \(\sigma \)-algebra of the completion and that this extension is unique. The key elements of this construction is the recursive character of the fractal structure as well as the use of the completion of the fractal structure.

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Acknowledgements

The authors would like to express their gratitude to anonymous reviewers whose suggestions, comments, and remarks have allowed them to improve the quality of this paper considerably.

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Authors and Affiliations

  1. Department of Mathematics, University of Almería, 04120, Almería, Spain

    J. F. Gálvez-Rodríguez & M. A. Sánchez-Granero

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  1. J. F. Gálvez-Rodríguez
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  2. M. A. Sánchez-Granero
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Correspondence to J. F. Gálvez-Rodríguez.

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M. A. Sánchez-Granero is supported by grants MTM2015-64373-P (MINECO/FEDER, UE) and PGC2018-101555-B-I00 (Ministerio Español de Ciencia, Innovación y Universidades and FEDER) and CDTIME.

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Gálvez-Rodríguez, J.F., Sánchez-Granero, M.A. Generating a Probability Measure on the Completion of a Fractal Structure. Results Math 74, 112 (2019). https://doi.org/10.1007/s00025-019-1039-2

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  • DOI: https://doi.org/10.1007/s00025-019-1039-2

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