# Lattice-valued Borel measures III

## Surjit Singh Khurana

Address: Department of Mathematics, University of Iowa Iowa City, Iowa 52242, USA

E-mail: khurana@math.uiowa.edu

Abstract: Let $X$ be a completely regular $T_{1}$ space, $E$ a boundedly complete vector lattice, $ C(X)$ $(C_{b}(X))$ the space of all (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 $E_{n}^{*}$ separates the points of $E$, an Alexanderovâ€™s type theorem is proved for a sequence of $\sigma $-additive measures.

AMSclassification: primary 28A33; secondary 28B15, 28C05, 28C15, 46G10, 46B42.

Keywords: order convergence, tight and \tau -smooth lattice-valued vector measures, measure representation of positive linear operators, Alexandrovâ€™s theorem.