Fubini’s Theorem for Non-Negative or Non-Positive Functions

Noboru Endou 1
  • 1 National Institute of Technology, Gifu College, , 2236-2, Gifu, Japan

Summary

The goal of this article is to show Fubini’s theorem for non-negative or non-positive measurable functions [], [], [], using the Mizar system [], []. We formalized Fubini’s theorem in our previous article [], but in that case we showed the Fubini’s theorem for measurable sets and it was not enough as the integral does not appear explicitly.

On the other hand, the theorems obtained in this paper are more general and it can be easily extended to a general integrable function. Furthermore, it also can be easy to extend to functional space Lp []. It should be mentioned also that Hölzl and Heller [] have introduced the Lebesgue integration theory in Isabelle/HOL and have proved Fubini’s theorem there.

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  • [1] 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.

  • [2] Heinz Bauer. Measure and Integration Theory. Walter de Gruyter Inc., 2002.

  • [3] Vladimir Igorevich Bogachev and Maria Aparecida Soares Ruas. Measure theory, volume 1. Springer, 2007.

  • [4] Noboru Endou. Product pre-measure. Formalized Mathematics, 24(1):69–79, 2016. doi:10.1515/forma-2016-0006.

  • [5] Noboru Endou. Fubini’s theorem on measure. Formalized Mathematics, 25(1):1–29, 2017. doi:10.1515/forma-2017-0001.

  • [6] Noboru Endou and Yasunari Shidama. Integral of measurable function. Formalized Mathematics, 14(2):53–70, 2006. doi:10.2478/v10037-006-0008-x.

  • [7] Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Basic properties of extended real numbers. Formalized Mathematics, 9(3):491–494, 2001.

  • [8] Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. The measurability of extended real valued functions. Formalized Mathematics, 9(3):525–529, 2001.

  • [9] Adam Grabowski, Artur Korniłowicz, and Adam Naumowicz. Four decades of Mizar. Journal of Automated Reasoning, 55(3):191–198, 2015. doi:10.1007/s10817-015-9345-1.

  • [10] P. R. Halmos. Measure Theory. Springer-Verlag, 1974.

  • [11] 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.

  • [12] Yasushige Watase, Noboru Endou, and Yasunari Shidama. On L 1 space formed by real-valued partial functions. Formalized Mathematics, 16(4):361–369, 2008. doi:10.2478/v10037-008-0044-9.

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