# Article

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Keywords:
order convergence; tight and $\tau$-smooth lattice-valued vector measures; measure representation of positive linear operators; Alexandrov’s theorem
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 (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 Alexanderov’s type theorem is proved for a sequence of $\sigma$-additive measures.
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