σ-ring and σ-algebra of Sets1

Noboru Endou; Kazuhisa Nakasho; Yasunari Shidama

Formalized Mathematics (2015)

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

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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