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Formalized Mathematics

(a computer assisted approach)

Editor-in-Chief: Matuszewski, Roman

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Volume 23, Issue 1 (Mar 2015)


σ-ring and σ-algebra of Sets1

Noboru Endou
  • Gifu National College of Technology, Gifu, Japan
/ Kazuhisa Nakasho
  • Shinshu University, Nagano, Japan
/ Yasunari Shidama
  • Shinshu University, Nagano, Japan
Published Online: 2015-03-31 | DOI: https://doi.org/10.2478/forma-2015-0004


In this article, semiring and semialgebra of sets are formalized so as to construct a measure of a given set in the next step. Although a semiring of sets has already been formalized in [13], that is, strictly speaking, a definition of a quasi semiring of sets suggested in the last few decades [15]. We adopt a classical definition of a semiring of sets here to avoid such a confusion. Ring of sets and algebra of sets have been formalized as non empty preboolean set [23] and field of subsets [18], respectively. In the second section, definitions of a ring and a σ-ring of sets, which are based on a semiring and a ring of sets respectively, are formalized and their related theorems are proved. In the third section, definitions of an algebra and a σ-algebra of sets, which are based on a semialgebra and an algebra of sets respectively, are formalized and their related theorems are proved. In the last section, mutual relationships between σ-ring and σ-algebra of sets are formalized and some related examples are given. The formalization is based on [15], and also referred to [9] and [16].

MSC: 03E30; 28A05; 03B35

Keywords: semiring of sets; σ-ring of sets; σ-algebra of sets

MML: identifier: SRINGS 3; version: 8.1.04 5.31.1231


  • [1] Grzegorz Bancerek. König’s theorem. Formalized Mathematics, 1(3):589–593, 1990.Google Scholar

  • [2] Grzegorz Bancerek. Tarski’s classes and ranks. Formalized Mathematics, 1(3):563–567, 1990.Google Scholar

  • [3] Grzegorz Bancerek. Continuous, stable, and linear maps of coherence spaces. Formalized Mathematics, 5(3):381–393, 1996.Google Scholar

  • [4] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.Google Scholar

  • [5] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.Google Scholar

  • [6] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.Google Scholar

  • [7] Józef Białas. The σ-additive measure theory. Formalized Mathematics, 2(2):263–270, 1991.Google Scholar

  • [8] Józef Białas. Properties of the intervals of real numbers. Formalized Mathematics, 3(2): 263–269, 1992.Google Scholar

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

  • [10] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55–65, 1990.Google Scholar

  • [11] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.Google Scholar

  • [12] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.Google Scholar

  • [13] Roland Coghetto. Semiring of sets. Formalized Mathematics, 22(1):79–84, 2014. doi:10.2478/forma-2014-0008.CrossrefGoogle Scholar

  • [14] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.Google Scholar

  • [15] D.F. Goguadze. About the notion of semiring of sets. Mathematical Notes, 74:346–351, 2003. ISSN 0001-4346. doi:10.1023/A:1026102701631.CrossrefGoogle Scholar

  • [16] P. R. Halmos. Measure Theory. Springer-Verlag, 1974.Google Scholar

  • [17] Jarosław Kotowicz and Konrad Raczkowski. Coherent space. Formalized Mathematics, 3 (2):255–261, 1992.Google Scholar

  • [18] Andrzej Nędzusiak. σ-fields and probability. Formalized Mathematics, 1(2):401–407, 1990.Google Scholar

  • [19] Andrzej Nędzusiak. Probability. Formalized Mathematics, 1(4):745–749, 1990.Google Scholar

  • [20] Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147–152, 1990.Google Scholar

  • [21] Konrad Raczkowski and Paweł Sadowski. Equivalence relations and classes of abstraction. Formalized Mathematics, 1(3):441–444, 1990.Google Scholar

  • [22] Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4): 341–347, 2003.Google Scholar

  • [23] Andrzej Trybulec and Agata Darmochwał. Boolean domains. Formalized Mathematics, 1 (1):187–190, 1990.Google Scholar

  • [24] Wojciech A. Trybulec. Non-contiguous substrings and one-to-one finite sequences. Formalized Mathematics, 1(3):569–573, 1990.Google Scholar

  • [25] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.Google Scholar

  • [26] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73–83, 1990.Google Scholar

  • [27] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181–186, 1990.Google Scholar

About the article

Received: 2015-02-18

Published Online: 2015-03-31

Published in Print: 2015-03-01

1This work was supported by JSPS KAKENHI 23500029 and 22300285.

Citation Information: Formalized Mathematics, ISSN (Online) 1898-9934, DOI: https://doi.org/10.2478/forma-2015-0004.

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© Noboru Endou et al.. This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 License. BY-SA 3.0

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