Lattice-valued Borel measures. III. (English) Zbl 1212.28009

Khurana, S. S.

Arch. Math., Brno 44, No. 4, 307-316 (2008).

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

Keywords:

order convergence; tight measure; \(\tau \)-smooth lattice-valued vector measure; measure representation; positive linear operator; Alexandrov’s theorem

Citations:

Zbl 0283.28010; Zbl 0325.28012