MATHEMATICA BOHEMICA, Vol. 133, No. 1, pp. 19-27 (2008)

Order convergence of vector measures on topological spaces

Surjit Singh Khurana

Surjit Singh Khurana, Department of Mathematics, University of Iowa, Iowa City, Iowa 52242, U.S.A., e-mail:

Abstract: 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's theorem about the convergent sequences of $\sigma$-additive measures is extended to the case of order convergence.

Keywords: order convergence, tight and $\tau$-smooth lattice-valued vector measures, measure representation of positive linear operators, Alexandrov's theorem

Classification (MSC2000): 28A33, 28B15, 28C05, 28C15, 46G10, 46B42

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