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.
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© 1996 Springer-Verlag New York, Inc.
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Simonnet, M. (1996). Measures on Semirings. In: Measures and Probabilities. Universitext. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-4012-9_2
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DOI: https://doi.org/10.1007/978-1-4612-4012-9_2
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