Let X be a completely regular Hausdorff space, E a boundedly complete vector lattice, Cb (X) the space of all, bounded, real-valued continuous functions on X, F the algebra generated by the zero-sets of X, and µ: Cb (X) → E a positive linear map. First we give a new proof that µ extends to a unique, finitely additive measure µ: F → E + such that ν 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 F are proved, which extend some known results. Also, under certain conditions, the well-known Alexandrov’s theorem about the convergent sequences of σ-additive measures is extended to the case of order convergence.
Suppose $E$ is an ordered locally convex space, $X_{1} $ and $X_{2} $ Hausdorff completely regular spaces and $Q$ a uniformly bounded, convex and closed subset of $ M_{t}^{+}(X_{1} \times X_{2}, E) $. For $ i=1,2 $, let $ \mu _{i} \in M_{t}^{+}(X_{i}, E) $. Then, under some topological and order conditions on $E$, necessary and sufficient conditions are established for the existence of an element in $Q$, having marginals $ \mu _{1} $ and $ \mu _{2}$.
A convergence structure generalizing the order convergence structure on the set of Hausdorff continuous interval functions is defined on the set of minimal usco maps. The properties of the obtained convergence space are investigated and essential links with the pointwise convergence and the order convergence are revealed. The convergence structure can be extended to a uniform convergence structure so that the convergence space is complete. The important issue of the denseness of the subset of all continuous functions is also addressed.