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σ-ring and σ-algebra of Sets1

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Endou N., Nakasho K., Shidama Y.

Formalized Mathematics

2015

tom 23

nr 1

51-57

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].

Ograniczanie wyników

1 Formalized Mathematics

1 Endou N.

1 Nakasho K.

1 Shidama Y.

1 2015

Wyniki wyszukiwania

σ-ring and σ-algebra of Sets1