# σ-ring and σ-algebra of Sets1

Formalized Mathematics (2015)

• Volume: 23, Issue: 1, page 51-57
• ISSN: 1426-2630

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## Abstract

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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].

## How to cite

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Noboru Endou, Kazuhisa Nakasho, and Yasunari Shidama. "σ-ring and σ-algebra of Sets1." Formalized Mathematics 23.1 (2015): 51-57. <http://eudml.org/doc/271085>.

@article{NoboruEndou2015,
abstract = {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].},
author = {Noboru Endou, Kazuhisa Nakasho, Yasunari Shidama},
journal = {Formalized Mathematics},
keywords = {semiring of sets; σ-ring of sets; σ-algebra of sets; -ring of sets; -algebra of sets},
language = {eng},
number = {1},
pages = {51-57},
title = {σ-ring and σ-algebra of Sets1},
url = {http://eudml.org/doc/271085},
volume = {23},
year = {2015},
}

TY - JOUR
AU - Noboru Endou
AU - Kazuhisa Nakasho
AU - Yasunari Shidama
TI - σ-ring and σ-algebra of Sets1
JO - Formalized Mathematics
PY - 2015
VL - 23
IS - 1
SP - 51
EP - 57
AB - 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].
LA - eng
KW - semiring of sets; σ-ring of sets; σ-algebra of sets; -ring of sets; -algebra of sets
UR - http://eudml.org/doc/271085
ER -

## References

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1. [1] Grzegorz Bancerek. König’s theorem. Formalized Mathematics, 1(3):589–593, 1990.
2. [2] Grzegorz Bancerek. Tarski’s classes and ranks. Formalized Mathematics, 1(3):563–567, 1990.
3. [3] Grzegorz Bancerek. Continuous, stable, and linear maps of coherence spaces. Formalized Mathematics, 5(3):381–393, 1996.
4. [4] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990. Zbl06213858
5. [5] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.
6. [6] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.
7. [7] Józef Białas. The σ-additive measure theory. Formalized Mathematics, 2(2):263–270, 1991.
8. [8] Józef Białas. Properties of the intervals of real numbers. Formalized Mathematics, 3(2): 263–269, 1992.
9. [9] Vladimir Igorevich Bogachev and Maria Aparecida Soares Ruas. Measure theory, volume 1. Springer, 2007.
10. [10] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55–65, 1990.
11. [11] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.
12. [12] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.
13. [13] Roland Coghetto. Semiring of sets. Formalized Mathematics, 22(1):79–84, 2014. doi:10.2478/forma-2014-0008. Zbl1298.28002
14. [14] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.
15. [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. Zbl1072.28001
16. [16] P. R. Halmos. Measure Theory. Springer-Verlag, 1974. Zbl0283.28001
17. [17] Jarosław Kotowicz and Konrad Raczkowski. Coherent space. Formalized Mathematics, 3 (2):255–261, 1992.
18. [18] Andrzej Nędzusiak. σ-fields and probability. Formalized Mathematics, 1(2):401–407, 1990.
19. [19] Andrzej Nędzusiak. Probability. Formalized Mathematics, 1(4):745–749, 1990.
20. [20] Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147–152, 1990.
21. [21] Konrad Raczkowski and Paweł Sadowski. Equivalence relations and classes of abstraction. Formalized Mathematics, 1(3):441–444, 1990.
22. [22] Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4): 341–347, 2003.
23. [23] Andrzej Trybulec and Agata Darmochwał. Boolean domains. Formalized Mathematics, 1 (1):187–190, 1990.
24. [24] Wojciech A. Trybulec. Non-contiguous substrings and one-to-one finite sequences. Formalized Mathematics, 1(3):569–573, 1990.
25. [25] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.
26. [26] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73–83, 1990.
27. [27] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181–186, 1990.

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