Multidimensional Measure Space and Integration

Noboru Endou; Yasunari Shidama

doi:10.2478/forma-2023-0017

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Multidimensional Measure Space and Integration

Formalized Mathematics

Volume 31 (2023): Issue 1 (September 2023)

By: Noboru Endou and Yasunari Shidama

Open Access

|Dec 2023

Abstract

This paper introduces multidimensional measure spaces and the integration of functions on these spaces in Mizar. Integrals on the multidimensional Cartesian product measure space are defined and appropriate formal apparatus to deal with this notion is provided as well.

References

Grzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pąk, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261–279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007/978-3-319-20615-8_17.
Search in Google Scholar Back to article
Vladimir Igorevich Bogachev and Maria Aparecida Soares Ruas. Measure theory, volume 1. Springer, 2007.
Search in Google Scholar Back to article
Sylvie Boldo, Catherine Lelay, and Guillaume Melquiond. Improving real analysis in Coq: A user-friendly approach to integrals and derivatives. In Chris Hawblitzel and Dale Miller, editors, Certified Programs and Proofs – Second International Conference, CPP 2012, Kyoto, Japan, December 13–15, 2012. Proceedings, volume 7679 of Lecture Notes in Computer Science, pages 289–304. Springer, 2012. doi:10.1007/978-3-642-35308-6_22.
Search in Google Scholar Back to article
Sylvie Boldo, Catherine Lelay, and Guillaume Melquiond. Formalization of real analysis: A survey of proof assistants and libraries. Mathematical Structures in Computer Science, 26:1196–1233, 2015.
Search in Google Scholar Back to article
Noboru Endou. Improper integral. Part II. Formalized Mathematics, 29(4):279–294, 2021. doi:10.2478/forma-2021-0024.
Search in Google Scholar Back to article
Noboru Endou. Fubini’s theorem on measure. Formalized Mathematics, 25(1):1–29, 2017. doi:10.1515/forma-2017-0001.
Search in Google Scholar Back to article
Noboru Endou. Fubini’s theorem. Formalized Mathematics, 27(1):67–74, 2019. doi:10.2478/forma-2019-0007.
Search in Google Scholar Back to article
Noboru Endou. Absolutely integrable functions. Formalized Mathematics, 30(1):31–51, 2022. doi:10.2478/forma-2022-0004.
Search in Google Scholar Back to article
Jacques D. Fleuriot. On the mechanization of real analysis in Isabelle/HOL. In Mark Aagaard and John Harrison, editors, Theorem Proving in Higher Order Logics, pages 145–161. Springer Berlin Heidelberg, 2000. ISBN 978-3-540-44659-0.
Search in Google Scholar Back to article
Ruben Gamboa. Continuity and Di erentiability, pages 301–315. Springer US, 2000. ISBN 978-1-4757-3188-0. doi:10.1007/978-1-4757-3188-0_18.
Search in Google Scholar Back to article
Adam Grabowski and Christoph Schwarzweller. Translating mathematical vernacular into knowledge repositories. In Michael Kohlhase, editor, Mathematical Knowledge Management, volume 3863 of Lecture Notes in Computer Science, pages 49–64. Springer, 2006. doi: https://doi.org/10.1007/11618027_4. 4th International Conference on Mathematical Knowledge Management, Bremen, Germany, MKM 2005, July 15–17, 2005, Revised Selected Papers.
Search in Google Scholar Back to article
Johannes Hölzl and Armin Heller. Three chapters of measure theory in Isabelle/HOL. In Marko C. J. D. van Eekelen, Herman Geuvers, Julien Schmaltz, and Freek Wiedijk, editors, Interactive Theorem Proving (ITP 2011), volume 6898 of LNCS, pages 135–151, 2011.
Search in Google Scholar Back to article
Hiroyuki Okazaki, Noboru Endou, and Yasunari Shidama. Cartesian products of family of real linear spaces. Formalized Mathematics, 19(1):51–59, 2011. doi:10.2478/v10037-011-0009-2.
Search in Google Scholar Back to article
M.M. Rao. Measure Theory and Integration. Marcel Dekker, 2nd edition, 2004.
Search in Google Scholar Back to article

Articles in this issue

DOI: https://doi.org/10.2478/forma-2023-0017 | Journal eISSN: 1898-9934 | Journal ISSN: 1426-2630

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Language: English

Page range: 181 - 192

Accepted on: Nov 21, 2023

Published on: Dec 31, 2023

Published by: University of Białystok

In partnership with: Paradigm Publishing Services

Publication frequency: 1 issue per year

Keywords:

measure in product spaces,

iterated integral

Related subjects:

Mathematics,

Logic and set theory,

Computer sciences,

Artificial intelligence

© 2023 Noboru Endou, Yasunari Shidama, published by University of Białystok
This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 License.

Volume 31 (2023): Issue 1 (September 2023)