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Generating a Probability Measure from a Fractal Structure

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Abstract

Fractal structures were introduced to characterize non-archimedean quasi-metrization, and there is a strong relationship between these two concepts. In this paper we show how to define a probability measure with the help of a fractal structure, by taking advantage of its recursive nature. One of the keys of this approach is the use of the completion of the fractal structure. Since we want to define a measure, we take into account the theorems on construction of outer measures (Method I and Method II). Once we have defined a first measure on the bicompletion of the space X, we explore conditions to ensure that the restriction of the measure to the original space is a probability measure. Finally, we prove that each probability measure on a space X can be constructed from a fractal structure by following the developed process.

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

    Present address: Department of Mathematics, University of Almería, 04120, Almería, Spain

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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 or M. A. Sánchez-Granero.

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    Miguel Ángel Sánchez-Granero acknowledges the support of grants PGC2018-101555-B-I00 (Ministerio Español de Ciencia, Innovación y Universidades and FEDER) and UAL18-FQM-B038-A (UAL/CECEU/FEDER) and CDTIME.

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    Gálvez-Rodríguez, J.F., Sánchez-Granero, M.A. Generating a Probability Measure from a Fractal Structure. Results Math 75, 101 (2020). https://doi.org/10.1007/s00025-020-01228-x

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