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A Formalization of the Lévy-Prokhorov Metric in Isabelle/HOL

Author Michikazu Hirata

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Author Details

Michikazu Hirata
  • School of Computing, Tokyo Institute of Technology, Japan

Funding

This work is supported by JSPS Research Fellowships for Young Scientists and JSPS KAKENHI Grant Number 23KJ0905.

Acknowledgements

I would like to thank Yasuhiko Minamide and Tetsuya Sato for commenting on a draft of this paper. I also thank anonymous reviewers for their helpful comments and suggestions.

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Michikazu Hirata. A Formalization of the Lévy-Prokhorov Metric in Isabelle/HOL. In 15th International Conference on Interactive Theorem Proving (ITP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 309, pp. 21:1-21:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ITP.2024.21

Abstract

The Lévy-Prokhorov metric is a metric between finite measures on a metric space. The metric was introduced to analyze weak convergence of measures. We formalize the Lévy-Prokhorov metric and prove Prokhorov’s theorem in Isabelle/HOL. Prokhorov’s theorem provides a condition for the relative compactness of sets of finite measures and plays essential roles in proofs of the central limit theorem, Sanov’s theorem in large deviation theory, and the existence of optimal coupling in transportation theory. Our formalization includes important results in mathematics such as the Riesz representation theorem, which is a theorem in functional analysis and used to prove Prokhorov’s theorem. We also apply the Lévy-Prokhorov metric to show that the measurable space of finite measures on a standard Borel space is again a standard Borel space.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Probability and statistics
  • Mathematics of computing → Mathematical analysis
Keywords
  • formalization of mathematics
  • measure theory
  • metric spaces
  • topology
  • Lévy-Prokhorov metric
  • Prokhorov’s theorem
  • Isabelle/HOL

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References

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