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}$.
Let $X$ be a completely regular Hausdorff space, $C_{b}(X)$ the space of all scalar-valued bounded continuous functions on $X$ with strict topologies. We prove that these are locally convex topological algebras with jointly continuous multiplication. Also we find the necessary and sufficient conditions for these algebras to be locally $m$-convex.