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].
@article{bwmeta1.element.doi-10_2478_forma-2015-0004,
author = {Noboru Endou and Kazuhisa Nakasho and Yasunari Shidama},
title = {$\sigma$-ring and $\sigma$-algebra of Sets1},
journal = {Formalized Mathematics},
volume = {23},
year = {2015},
pages = {51-57},
zbl = {1317.28001},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_forma-2015-0004}
}
Noboru Endou; Kazuhisa Nakasho; Yasunari Shidama. σ-ring and σ-algebra of Sets1. Formalized Mathematics, Tome 23 (2015) pp. 51-57. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_forma-2015-0004/
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