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Integration with respect to finitely additive measures. (English) Zbl 0771.28004

Positive operators, Riesz spaces, and economics, Proc. Conf., Pasadena/CA (USA) 1990, Stud. Econ. Theory 2, 109-150 (1991).
[For the entire collection see Zbl 0746.00055.]
Throughout, a measure is a finitely additive real-valued function of bounded variation whose domain is an algebra of sets. This paper is an investigation of integration of (mainly) real-valued functions with respect to such measures viewed from the perspective of Riesz spaces. Specific topics addressed include: extensions of measures including a characterization of the algebra of measurable sets determined by the outer measure of a measure; a Riemann integral for measures and a characterization of Riemann integrable functions; the Dunford integral, the manner in which it extends the Riemann integral, and the Beppo Levi theorem with applications to a functional calculus; the set \(A_ \mu\) of \(\mu\)-absolutely continuous measures; the Radon-Nikodým theorem, first abstractly in the context of Kakutani’s \(L\) and \(M\) spaces, then for arbitrary \(\mu\)-absolutely continuous measures in which case the density is in general an order continuous linear functional defined on an order dense ideal of \(A_ \mu\) which is \(\mu\)-integrable in a refinement sense; \({\mathcal L}^ p\) spaces; and necessary and sufficient conditions for the norm-completeness of \({\mathcal L}^ 1\). This is a very full paper drawing many connections and parallels between various integrals and approaches to integration involving bounded finitely additive measures.

MSC:

28A25 Integration with respect to measures and other set functions
46A40 Ordered topological linear spaces, vector lattices
28A15 Abstract differentiation theory, differentiation of set functions
46E27 Spaces of measures
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

Citations:

Zbl 0746.00055
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