Fubini’s Theorem on Measure

The purpose of this article is to show Fubini’s theorem on measure [16], [4], [7], [15], [18]. Some theorems have the possibility of slight generalization, but we have priority to avoid the complexity of the description. First of all, for the product measure constructed in [14], we show some theorems. Then we introduce the section which plays an important role in Fubini’s theorem, and prove the relevant proposition. Finally we show Fubini’s theorem on measure.National Institute of Technology, Gifu College 2236-2 Kamimakuwa, Motosu, Gifu, JapanGrzegorz Bancerek. Curried and uncurried functions. Formalized Mathematics, 1(3): 537-541, 1990.Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Heinz Bauer. Measure and Integration Theory. Walter de Gruyter Inc., 2002.Józef Białas. The σ-additive measure theory. Formalized Mathematics, 2(2):263-270, 1991.Józef Białas. Series of positive real numbers. Measure theory. Formalized Mathematics, 2(1):173-183, 1991.Vladimir Igorevich Bogachev and Maria Aparecida Soares Ruas. Measure theory, volume 1. Springer, 2007.Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.Czesław Byliński. Some properties of restrictions of finite sequences. Formalized Mathematics, 5(2):241-245, 1996.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Noboru Endou. Product pre-measure. Formalized Mathematics, 24(1):69-79, 2016.Gerald B. Folland. Real Analysis: Modern Techniques and Their Applications. Wiley, 2nd edition, 1999.P. R. Halmos. Measure Theory. Springer-Verlag, 1974.Andrzej Nędzusiak. σ-fields and probability. Formalized Mathematics, 1(2):401-407, 1990.M.M. Rao. Measure Theory and Integration. Marcel Dekker, 2nd edition, 2004.Wojciech A. Trybulec. Non-contiguous substrings and one-to-one finite sequences. Formalized Mathematics, 1(3):569-573, 1990.Hiroshi Yamazaki, Noboru Endou, Yasunari Shidama, and Hiroyuki Okazaki. Inferior limit, superior limit and convergence of sequences of extended real numbers. Formalized Mathematics, 15(4):231-236, 2007. doi: 10.2478/v10037-007-0026-3.Bo Zhang, Hiroshi Yamazaki, and Yatsuka Nakamura. Limit of sequence of subsets. Formalized Mathematics, 13(2):347-352, 2005.Bo Zhang, Hiroshi Yamazaki, and Yatsuka Nakamura. Some equations related to the limit of sequence of subsets. Formalized Mathematics, 13(3):407-412, 2005

Product Pre-Measure

In this article we formalize in Mizar [5] product pre-measure on product sets of measurable sets. Although there are some approaches to construct product measure [22], [6], [9], [21], [25], we start it from σ-measure because existence of σ-measure on any semialgebras has been proved in [15]. In this approach, we use some theorems for integrals.EndouGifu Noboru - Gifu National College of Technology Gifu, JapanGrzegorz Bancerek. Towards the construction of a model of Mizar concepts. Formalized Mathematics, 16(2):207-230, 2008. doi:10.2478/v10037-008-0027-x.Grzegorz Bancerek. Curried and uncurried functions. Formalized Mathematics, 1(3): 537-541, 1990.Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.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.Heinz Bauer. Measure and Integration Theory. Walter de Gruyter Inc.Józef Białas. The σ-additive measure theory. Formalized Mathematics, 2(2):263-270, 1991.Józef Białas. Series of positive real numbers. Measure theory. Formalized Mathematics, 2(1):173-183, 1991.Vladimir Igorevich Bogachev and Maria Aparecida Soares Ruas. Measure theory, volume 1. Springer, 2007.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Byliński. Basic functions and operations on functions. Formalized Mathematics, 1(1):245-254, 1990.Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Noboru Endou. Construction of measure from semialgebra of sets. Formalized Mathematics, 23(4):309-323, 2015. doi:10.1515/forma-2015-0025.Noboru Endou and Yasunari Shidama. Integral of measurable function. Formalized Mathematics, 14(2):53-70, 2006. doi:10.2478/v10037-006-0008-x.Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Definitions and basic properties of measurable functions. Formalized Mathematics, 9(3):495-500, 2001.Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. The measurability of extended real valued functions. Formalized Mathematics, 9(3):525-529, 2001.Noboru Endou, Keiko Narita, and Yasunari Shidama. The Lebesgue monotone convergence theorem. Formalized Mathematics, 16(2):167-175, 2008. doi:10.2478/v10037-008-0023-1.Noboru Endou, Hiroyuki Okazaki, and Yasunari Shidama. Hopf extension theorem of measure. Formalized Mathematics, 17(2):157-162, 2009. doi:10.2478/v10037-009-0018-6.Gerald B. Folland. Real Analysis: Modern Techniques and Their Applications. Wiley, 2 edition, 1999.P. R. Halmos. Measure Theory. Springer-Verlag, 1974.Andrzej Nędzusiak. σ-fields and probability. Formalized Mathematics, 1(2):401-407, 1990.Beata Perkowska. Functional sequence from a domain to a domain. Formalized Mathematics, 3(1):17-21, 1992.M.M. Rao. Measure Theory and Integration. Marcel Dekker, 2nd edition, 2004.Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1 (2):329-334, 1990.Wojciech A. Trybulec. Non-contiguous substrings and one-to-one finite sequences. Formalized Mathematics, 1(3):569-573, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.Hiroshi Yamazaki, Noboru Endou, Yasunari Shidama, and Hiroyuki Okazaki. Inferior limit, superior limit and convergence of sequences of extended real numbers. Formalized Mathematics, 15(4):231-236, 2007. doi:10.2478/v10037-007-0026-3

Double Series and Sums

In this paper the author constructs several properties for double series and its convergence. The notions of convergence of double sequence have already been introduced in our previous paper [18]. In section 1 we introduce double series and their convergence. Then we show the relationship between Pringsheim-type convergence and iterated convergence. In section 2 we study double series having non-negative terms. As a result, we have equality of three type sums of non-negative double sequence. In section 3 we show that if a non-negative sequence is summable, then the sequence of rearrangement of terms is summable and it has the same sums. In the last section two basic relations between double sequences and matrices are introduced.This work was supported by JSPS KAKENHI 23500029.Gifu National College of Technology Gifu, JapanGrzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.Grzegorz Bancerek. Tarski’s classes and ranks. Formalized Mathematics, 1(3):563-567, 1990.Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Grzegorz Bancerek. Representation theorem for stacks. Formalized Mathematics, 19(4): 241-250, 2011. doi:10.2478/v10037-011-0033-2.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Czesław Bylinski. Binary operations. Formalized Mathematics, 1(1):175-180, 1990.Czesław Bylinski. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.Czesław Bylinski. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.Czesław Bylinski. Some properties of restrictions of finite sequences. Formalized Mathematics, 5(2):241-245, 1996.Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Bylinski. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661-668, 1990.Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.Noboru Endou, Keiko Narita, and Yasunari Shidama. The Lebesgue monotone convergence theorem. Formalized Mathematics, 16(2):167-175, 2008. doi:10.2478/v10037-008-0023-1.Noboru Endou, Hiroyuki Okazaki, and Yasunari Shidama. Double sequences and limits. Formalized Mathematics, 21(3):163-170, 2013. doi:10.2478/forma-2013-0018.Fuguo Ge and Xiquan Liang. On the partial product of series and related basic inequalities. Formalized Mathematics, 13(3):413-416, 2005.Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5): 841-845, 1990.Artur Korniłowicz. On the real valued functions. Formalized Mathematics, 13(1):181-187, 2005.Jarosław Kotowicz. Convergent sequences and the limit of sequences. Formalized Mathematics, 1(2):273-275, 1990.Gilbert Lee. Weighted and labeled graphs. Formalized Mathematics, 13(2):279-293, 2005.Konrad Raczkowski and Andrzej Nedzusiak. Series. Formalized Mathematics, 2(4):449-452, 1991.Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.Wojciech A. Trybulec. Non-contiguous substrings and one-to-one finite sequences. Formalized Mathematics, 1(3):569-573, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.Bo Zhang and Yatsuka Nakamura. The definition of finite sequences and matrices of probability, and addition of matrices of real elements. Formalized Mathematics, 14(3): 101-108, 2006. doi:10.2478/v10037-006-0012-1

Extended Real-Valued Double Sequence and Its Convergence

AbstractIn this article we introduce the convergence of extended realvalued double sequences [16], [17]. It is similar to our previous articles [15], [10]. In addition, we also prove Fatou’s lemma and the monotone convergence theorem for double sequences.This work was supported by JSPS KAKENHI 2350002Gifu National College of Technology, Gifu, JapanGrzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Józef Białas. Infimum and supremum of the set of real numbers. Measure theory. Formalized Mathematics, 2(1):163-171, 1991.Józef Białas. Series of positive real numbers. Measure theory. Formalized Mathematics, 2(1):173-183, 1991.Czesław Bylinski. Binary operations. Formalized Mathematics, 1(1):175-180, 1990.Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Noboru Endou. Double series and sums. Formalized Mathematics, 22(1):57-68, 2014. doi:10.2478/forma-2014-0006. [Crossref]Noboru Endou and Yasunari Shidama. Integral of measurable function. Formalized Mathematics, 14(2):53-70, 2006. doi:10.2478/v10037-006-0008-x. [Crossref]Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Basic properties of extended real numbers. Formalized Mathematics, 9(3):491-494, 2001.Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Definitions and basic properties of measurable functions. Formalized Mathematics, 9(3):495-500, 2001.Noboru Endou, Keiko Narita, and Yasunari Shidama. The Lebesgue monotone convergence theorem. Formalized Mathematics, 16(2):167-175, 2008. doi:10.2478/v10037-008-0023-1. [Crossref]Noboru Endou, Hiroyuki Okazaki, and Yasunari Shidama. Double sequences and limits. Formalized Mathematics, 21(3):163-170, 2013. doi:10.2478/forma-2013-0018. [Crossref]Gerald B. Folland. Real Analysis: Modern Techniques and Their Applications. Wiley, 2 edition, 1999.D.J.H. Garling. A Course in Mathematical Analysis: Volume 1, Foundations and Elementary Real Analysis, volume 1. Cambridge University Press, 2013.Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5): 841-845, 1990.Jarosław Kotowicz. Monotone real sequences. Subsequences. Formalized Mathematics, 1 (3):471-475, 1990.Jarosław Kotowicz. Convergent sequences and the limit of sequences. Formalized Mathematics, 1(2):273-275, 1990.Adam Naumowicz. Conjugate sequences, bounded complex sequences and convergent complex sequences. Formalized Mathematics, 6(2):265-268, 1997.Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.Hiroshi Yamazaki, Noboru Endou, Yasunari Shidama, and Hiroyuki Okazaki. Inferior limit, superior limit and convergence of sequences of extended real numbers. Formalized Mathematics, 15(4):231-236, 2007. doi:10.2478/v10037-007-0026-3. [Crossref

Improper Integral. Part II

In this article, using the Mizar system [2], [3], we deal with Riemann’s improper integral on infinite interval [1]. As with [4], we referred to [6], which discusses improper integrals of finite values.National Institute of Technology, Gifu College, 2236-2 Kamimakuwa, Motosu, Gifu, JapanTom M. Apostol. Mathematical Analysis. Addison-Wesley, 1969.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:10.1007/978-3-319-20615-817.Grzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, and Karol Pąk. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9–32, 2018. doi:10.1007/s10817-017-9440-6.Noboru Endou. Improper integral. Part I. Formalized Mathematics, 29(4):201–220, 2021. doi:10.2478/forma-2021-0019.Noboru Endou, Yasunari Shidama, and Masahiko Yamazaki. Integrability and the integral of partial functions from R into R. Formalized Mathematics, 14(4):207–212, 2006. doi:10.2478/v10037-006-0023-y.Masahiko Yamazaki, Hiroshi Yamazaki, and Yasunari Shidama. Extended Riemann integral of functions of real variable and one-sided Laplace transform. Formalized Mathematics, 16(4):311–317, 2008. doi:10.2478/v10037-008-0038-7.29427929

Fubini’s Theorem

Fubini theorem is an essential tool for the analysis of high-dimensional space [8], [2], [3], a theorem about the multiple integral and iterated integral. The author has been working on formalizing Fubini’s theorem over the past few years [4], [6] in the Mizar system [7], [1]. As a result, Fubini’s theorem (30) was proved in complete form by this article.National Institute of Technology, Gifu College, 2236-2 Kamimakuwa, Motosu, Gifu, JapanGrzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, and Karol Pak. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9–32, 2018. doi:10.1007/s10817-017-9440-6.Heinz Bauer. Measure and Integration Theory. Walter de Gruyter Inc., 2002.Vladimir Igorevich Bogachev and Maria Aparecida Soares Ruas. Measure theory, volume 1. Springer, 2007.Noboru Endou. Fubini’s theorem on measure. Formalized Mathematics, 25(1):1–29, 2017. doi:10.1515/forma-2017-0001.Noboru Endou. Integral of non positive functions. Formalized Mathematics, 25(3):227–240, 2017. doi:10.1515/forma-2017-0022.Noboru Endou. Fubini’s theorem for non-negative or non-positive functions. Formalized Mathematics, 26(1):49–67, 2018. doi:10.2478/forma-2018-0005.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.P. R. Halmos. Measure Theory. Springer-Verlag, 1974.271677

Relationship between the Riemann and Lebesgue Integrals

The goal of this article is to clarify the relationship between Riemann and Lebesgue integrals. In previous article [5], we constructed a onedimensional Lebesgue measure. The one-dimensional Lebesgue measure provides a measure of any intervals, which can be used to prove the well-known relationship [6] between the Riemann and Lebesgue integrals [1]. We also proved the relationship between the integral of a given measure and that of its complete measure. As the result of this work, the Lebesgue integral of a bounded real valued function in the Mizar system [2], [3] can be calculated by the Riemann integral.National Institute of Technology, Gifu College, 2236-2 Kamimakuwa, Motosu, Gifu, JapanTom M. Apostol. Mathematical Analysis. Addison-Wesley, 1969.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:10.1007/978-3-319-20615-817.Grzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, and Karol Pąk. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9–32, 2018. doi:10.1007/s10817-017-9440-6.Noboru Endou. Product pre-measure. Formalized Mathematics, 24(1):69–79, 2016. doi:10.1515/forma-2016-0006.Noboru Endou. Reconstruction of the one-dimensional Lebesgue measure. Formalized Mathematics, 28(1):93–104, 2020. doi:10.2478/forma-2020-0008.Gerald B. Folland. Real Analysis: Modern Techniques and Their Applications. Wiley, 2nd edition, 1999.Hiroshi Yamazaki, Noboru Endou, Yasunari Shidama, and Hiroyuki Okazaki. Inferior limit, superior limit and convergence of sequences of extended real numbers. Formalized Mathematics, 15(4):231–236, 2007. doi:10.2478/v10037-007-0026-3.29418519

Absolutely Integrable Functions

The goal of this article is to clarify the relationship between Riemann’s improper integrals and Lebesgue integrals. In previous articles [6], [7], we treated Riemann’s improper integrals [1], [11] and [4] on arbitrary intervals. Therefore, in this article, we will continue to clarify the relationship between improper integrals and Lebesgue integrals [8], using the Mizar [3], [2] formalism.Noboru Endou - National Institute of Technology, Gifu College, 2236-2 Kamimakuwa, Motosu, Gifu, JapanTom M. Apostol. Mathematical Analysis. Addison-Wesley, 1969.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.Grzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, and Karol Pąk. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9–32, 2018. doi:10.1007/s10817-017-9440-6.Vladimir Igorevich Bogachev and Maria Aparecida Soares Ruas. Measure theory, volume 1. Springer, 2007.Noboru Endou. Extended real-valued double sequence and its convergence. Formalized Mathematics, 23(3):253–277, 2015. doi:10.1515/forma-2015-0021.Noboru Endou. Improper integral. Part I. Formalized Mathematics, 29(4):201–220, 2021. doi:10.2478/forma-2021-0019.Noboru Endou. Improper integral. Part II. Formalized Mathematics, 29(4):279–294, 2021. doi:10.2478/forma-2021-0024.Noboru Endou. Relationship between the Riemann and Lebesgue integrals. Formalized Mathematics, 29(4):185–199, 2021. doi:10.2478/forma-2021-0018.Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Basic properties of extended real numbers. Formalized Mathematics, 9(3):491–494, 2001.Noboru Endou, Yasunari Shidama, and Masahiko Yamazaki. Integrability and the integral of partial functions from R into R. Formalized Mathematics, 14(4):207–212, 2006. doi:10.2478/v10037-006-0023-yGerald B. Folland. Real Analysis: Modern Techniques and Their Applications. Wiley, 2nd edition, 1999.301315

Improper Integral. Part I

In this article, we deal with Riemann’s improper integral [1], using the Mizar system [2], [3]. Improper integrals with finite values are discussed in [5] by Yamazaki et al., but in general, improper integrals do not assume that they are finite. Therefore, we have formalized general improper integrals that does not limit the integral value to a finite value. In addition, each theorem in [5] assumes that the domain of integrand includes a closed interval, but since the improper integral should be discusses based on the half-open interval, we also corrected it.National Institute of Technology, Gifu College, 2236-2 Kamimakuwa, Motosu, Gifu, JapanTom M. Apostol. Mathematical Analysis. Addison-Wesley, 1969.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:10.1007/978-3-319-20615-817.Grzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, and Karol Pąk. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9–32, 2018. doi:10.1007/s10817-017-9440-6.Noboru Endou, Yasunari Shidama, and Masahiko Yamazaki. Integrability and the integral of partial functions from R into R. Formalized Mathematics, 14(4):207–212, 2006. doi:10.2478/v10037-006-0023-y.Masahiko Yamazaki, Hiroshi Yamazaki, and Yasunari Shidama. Extended Riemann integral of functions of real variable and one-sided Laplace transform. Formalized Mathematics, 16(4):311–317, 2008. doi:10.2478/v10037-008-0038-7.29420122