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Lattice-valued Borel measures. III. (English) Zbl 1212.28009
Summary: Let \(X\) be a completely regular \(T_{1}\) space, \(E\) a boundedly complete vector lattice, \( C(X)\) \((C_{b}(X))\) the space of all (bounded), real-valued continuous functions on \(X\). In order convergence, we consider \(E\)-valued, order-bounded, \(\sigma \)-additive, \(\tau \)-additive, and tight measures on \(X\) and prove some order-theoretic and topological properties of these measures. Also, for an order-bounded, \(E\)-valued (for some special \(E\)) linear map on \(C(X)\), a measure representation result is proved. In case \(E_{n}^{\ast }\) separates the points of \(E\), an Alexanderov’s type theorem is proved for a sequence of \(\sigma \)-additive measures.
[For part I, see Rocky Mt. J. Math. 6, 377–382 (1976; Zbl 0283.28010), for part II, Trans. Am. Math. Soc. 235, 205–211 (1978; Zbl 0325.28012).]
MSC:
28A33 Spaces of measures, convergence of measures
28B15 Set functions, measures and integrals with values in ordered spaces
28C05 Integration theory via linear functionals (Radon measures, Daniell integrals, etc.), representing set functions and measures
28C15 Set functions and measures on topological spaces (regularity of measures, etc.)
46G10 Vector-valued measures and integration
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