Integrable Functions for Measures on Semirings

Summary

6.1 Let µ be a measure on S semiring in Ω. A ⊂ Ω H is µ-measurable if and only if, for every E ∈ S, the set A ∩ E is integrable (Proposition 6.1.3). A function from Ω, into some metrizable space is µ-measurable if and only if, for every E ∈ S, the restriction of f to E is the limit a.e. of simple mappings (Proposition 6.1.4). Finally, we give two other important necessary and sufficient conditions for f to be µ-measurable.

6.2 Let F be a Banach space with dual F′. If f ∈ ℒ ¹ _F (µ), then N₁(f) = sup_g∈B |∫ fgdµ| where B is the closed unit ball of St(S, F′) (Theorem 6.2.1).

6.3 This section generalizes the following example. Let S be the semiring of finite subsets of N. Then E ↦ card E, where card E is the cardinality of E, is a measure on S, called the counting measure. It is defined by the unit mass at each point. Such a measure is atomic (cf. Section 3.8).

6.4 In this section we return to the study of prolongations of a measure (cf. Section 3.9).