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Formalized Mathematics

(a computer assisted approach)

Editor-in-Chief: Matuszewski, Roman

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Volume 23, Issue 1


σ-ring and σ-algebra of Sets1

Noboru Endou / Kazuhisa Nakasho / Yasunari Shidama
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


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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, Volume 23, Issue 1, Pages 51–57, 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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