Measures on Semirings

Summary

2.1 Given a nonempty set Ω, the power set of Ω, equipped with symmetric difference and intersection, is a ring. A nonempty subring is called a ring. A σ-ring is a ring which is stable under countable unions. The subset S of semiclosed subintervals [a, b] is not a ring but merely a semiring (Definition 2.1.2). The σ-ring of Halmos sets and the σ-algebra of Borel sets (Definition 2.1.6), in a topological space Ω, are among the most important examples of σ-rings and algebras and will be widely used throughout this book.

2.2 In this section we first define quasi-measures, then measures on semirings. A function μ on a semiring S is a measure if and only if, for any sequence of disjoints sets A _i ∈ S such that ⋃ A _i ⊂ A ∈ S, the series $\sum {\mu \left( {{A_i}} \right)}$ converges, and if $\mu (A) = \sum {\mu \left( {{A_i}} \right)}$ whenever A = ⋃ A _i (Theorem 2.2.3). Any complex measure on a σ-ring is bounded (Proposition 2.2.3). Theorem 2.2.5 gives a basic relationship between abstract measures defined on a semiring S and Daniell measures defined on S-simple functions (“step functions”).

2.3 We introduce Lebesgue measure as a measure defined on the semiring of all semiclosed intervals [α, β] included in some interval I.