σ-ring and σ-algebra of Sets1

In this article, semi-ring and semi-algebra of sets are formalized so as to construct a measure of a given set in the next step. Although a semi-ring of sets has already been formalized in [12], that is, strictly speaking, a definition of a quasi semi-ring of sets suggested in the last few decades [14]. We adopt a classical definition of a semi-ring of sets here to avoid such a confusion. Ring of sets and algebra of sets have been formalized as non empty preBoolean set [21] and Field_Subset [16], respectively. In the second section, definitions of a ring and a σ-ring of sets, which are based on a semi-ring 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 semi-algebra 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.