# 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

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topNoboru 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 -

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