Order convergence of vector measures on topological spaces. (English) Zbl 1199.28008

Khurana, Surjit Singh

Math. Bohem. 133, No. 1, 19-27 (2008).

Summary: Let \(X\) be a completely regular Hausdorff space, \(E\) a boundedly complete vector lattice, \(C_{b}(X)\) the space of all, bounded, real-valued continuous functions on \(X\), \(\mathcal {F}\) the algebra generated by the zero-sets of \(X\), and \(\mu \: C_{b}(X) \to E\) a positive linear map. First, we give a new proof that \(\mu \) extends to a unique, finitely additive measure \(\mu \: \mathcal {F} \to E^{+}\) such that \(\nu \) is inner regular by zero-sets and outer regular by cozero sets. Then, some order-convergence theorems about nets of \(E^{+}\)-valued finitely additive measures on \(\mathcal {F}\) are proved, which extend some known results. Also, under certain conditions, the well-known Alexandrov theorem about convergent sequences of \(\sigma \)-additive measures is extended to the case of order convergence.

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 and \(\tau \)-smooth lattice-valued vector measures; measure representation of positive linear operators; Alexandrov’s theorem